Inclusion Criteria:

          -  Written informed consent and HIPAA authorization for release of personal health
             information prior to registration. NOTE: HIPAA authorization may be included in the
             informed consent or obtained separately.

          -  Age ≥ 18 years at the time of consent.

          -  ECOG Performance Status (PS) of 0-1within 28 days prior to registration. NOTE: Within
             0-3 days prior to the anticipated C1D1, ECOG PS must be 0-1.

          -  Archived tumor tissue must be available and identified during screening and shipped
             after subject registration. If archived tissue is not available and the subject is not
             undergoing a standard of care biopsy, the subject is not eligible for trial
             participation.

          -  PD-L1 results are required, if available. If PD-L1 testing has not been done, it
             should be ordered as standard of care prior to C1D1. PD-L1 testing must be performed
             by a CLIA certified lab using the Dako 22C3 antibody.

          -  Histologically or cytologically proven adenocarcinoma of the stomach or GEJ.

          -  Metastatic, recurrent, or locally advanced unresectable disease.

          -  Intolerant of or progressed on at least one prior line of therapy for metastatic
             disease. NOTE: Neoadjuvant or adjuvant therapy administered < 12 months prior to
             diagnosis of recurrent disease counts as one line of therapy.

          -  Measurable disease per RECIST v1.1

          -  Candidate for pembrolizumab, ramucirumab, and paclitaxel

          -  Demonstrate adequate organ function as defined in the table below. All screening labs
             to be obtained within 28 days prior to registration. NOTE: Labs must also be obtained
             within 10 days prior to C1D1 treatment.

               -  Absolute Neutrophil Count (ANC) ≥ 1,100/mm3

               -  Hemoglobin (Hgb) ≥ 8.5 g/dL without transfusion or EPO dependency

               -  Platelets ≥ 100,000 / mcL

               -  Creatinine OR Calculated creatinine clearance1 ≤ 1.5 x upper limit of normal
                  (ULN) OR ≥ 60 mL/min for subject with creatinine levels > 1.5 x institutional ULN

               -  Total bilirubin ≤ 1.5 X ULN OR Direct bilirubin ≤ ULN for subjects with total
                  bilirubin levels > 1.5 ULN OR total bilirubin ≤ 2 x ULN if liver metastases are
                  present (patients with Gilbert's syndrome are allowed)

               -  Aspartate aminotransferase (AST) and Alanine aminotransferase (ALT) ≤ 2.5 X ULN
                  OR ≤ 5 x ULN for subjects with liver metastases

               -  Albumin > 2.5 g/dL

          -  Willingness to provide tissue and blood samples for correlative research purposes.
             NOTE: Enrollment in parallel biopsy protocol, if open for enrollment, is required.
             Parallel biopsy protocol entitled: "Exploration of tumor biology in patients with
             metastatic esophageal and gastric cancer (biorepository protocol for prospective
             tissue collection)".

          -  Females of childbearing potential must have a negative pregnancy test within 72 hours
             prior to registration. If the urine test is positive or cannot be confirmed as
             negative, a serum pregnancy test will be required. NOTE: Females are considered of
             child bearing potential unless they are surgically sterile (have undergone a
             hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are
             naturally postmenopausal for at least 12 consecutive months

          -  Females of childbearing potential and males must be willing to abstain from
             heterosexual intercourse or to use 2 forms of effective methods of contraception from
             the time of informed consent (females)/prior to C1D1 (males) until 120 days after
             treatment discontinuation. See the protocol for contraception options.

          -  Willingness to return to the enrolling institution for follow up

        Exclusion Criteria:

          -  Past medical history of interstitial lung disease (ILD), drug-induced ILD, radiation
             pneumonitis which required steroid treatment, or any evidence of clinically active
             interstitial lung disease (relevant for ramucirumab).

          -  Any of the following cardiac criteria (relevant for ramucirumab):

               -  Mean resting corrected QT interval (QTc using Fridericia's formula) > 470 msec.

               -  Any clinically important abnormalities in rhythm, conduction or morphology of
                  resting ECG e.g., complete left bundle branch block, third degree heart block,
                  second degree heart block, PR interval >250msec.

               -  Symptomatic heart failure, uncontrolled hypokalemia despite repletion, congenital
                  long QT syndrome, family history of long QT syndrome or unexplained sudden death
                  under 40 years of age in first degree relatives. NOTE: Factors that increase risk
                  of QTc prolongation or risk of arrhythmia, such as concomitant medications, may
                  require increased monitoring during Combination Therapy (See Section 7).

               -  The patient has experienced any arterial thromboembolic events, including but not
                  limited to myocardial infarction, transient ischemic attack, cerebrovascular
                  accident, or unstable angina, within 6 months prior to first dose of protocol
                  therapy.

               -  The patient has uncontrolled or poorly-controlled hypertension (>160 mmHg
                  systolic or > 100 mmHg diastolic for >4 weeks) despite standard medical
                  management.

          -  The patient has experienced any Grade 3-4 GI bleeding within 3 months prior to first
             dose of protocol therapy (relevant for ramucirumab).

          -  Cirrhosis at a level of Child-Pugh B (or worse) or cirrhosis (any degree) and a
             history of hepatic encephalopathy or clinically meaningful ascites resulting from
             cirrhosis. Clinically meaningful ascites is defined as ascites from cirrhosis
             requiring diuretics or paracentesis (relevant for ramucirumab).

          -  Any hemoptysis (defined as bright red blood or ≥ 1/2 teaspoon) within 2 months prior
             to first dose of protocol therapy or with radiographic evidence of intratumor
             cavitation or has radiologically documented evidence of major blood vessel invasion or
             encasement by cancer (relevant for ramucirumab).

          -  The patient has a prior history of GI perforation/fistula (within 6 months of first
             dose of protocol therapy) or risk factors for perforation (relevant for ramucirumab).

          -  The patient has a serious or nonhealing wound, ulcer, or bone fracture within 28 days
             prior to first dose of protocol therapy (relevant for ramucirumab).

          -  The patient has undergone major surgery within 28 days prior to first dose of protocol
             therapy prior to the first dose of protocol therapy. The patient has elective or
             planned major surgery to be performed during the course of the clinical trial
             (relevant for ramucirumab).

          -  Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment
             of the investigator, would make the patient inappropriate for entry into this study or
             interfere significantly with the proper assessment of safety and toxicity of the
             prescribed regimens.

          -  Has active autoimmune disease that has required systemic treatment in the past 2 years
             (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive
             drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid
             replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a
             form of systemic treatment. NOTE: Inhaled steroids or steroid injections for joint
             disease are allowed.

          -  Is currently participating and receiving study therapy or has participated in a study
             of an investigational agent and received study therapy or used an investigational
             device within 4 weeks of the first dose of treatment.

          -  Disease progression according to RECIST v1.1 or irRECIST, or treatment intolerance,
             during prior therapy with an anti-PD-1, anti-PD-L1, or anti- PD-L2 agent. NOTE: Stable
             or responsive disease on anti-PD-1/L1-L2 therapy (without concurrent cytotoxic
             therapy) is allowed.

          -  Weight loss ≥5% during prior anti-PD-1, anti-PD-L1, or anti- PD-L2-containing therapy
             (from time of initiation of such therapy to most recent dose).

          -  Prior therapy combining anti-angiogenesis agent with cytotoxic agent(s). NOTE: Prior
             single-agent anti-angiogenesis therapy (eg, ramucirumab monotherapy) is allowed.

          -  Patients known to be HIV positive.

          -  Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA
             [qualitative] is detected).

          -  Prior severe allergic reactions to a monoclonal antibody or hypersensitivity to
             pembrolizumab or any of its excipients.

          -  Has had a prior anti-cancer monoclonal antibody (mAb) within 3 weeks prior to study
             Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events
             due to agents administered more than 3 weeks earlier.

          -  Other active malignancy which requires current treatment and which in the opinion of
             the site investigator is likely to interfere with evaluation of disease assessment.
             NOTE: Continuation of hormonal therapies is allowed.

          -  Patients with known active central nervous system (CNS) metastases may participate
             provided they are stable (without evidence of progression by imaging for at least four
             weeks prior to the first dose of trial treatment and any neurologic symptoms have
             returned to baseline), have no evidence of new or enlarging brain metastases, and are
             not using steroids for at least 7 days prior to trial treatment. This exception does
             not include carcinomatous meningitis which is excluded regardless of clinical
             stability.

          -  Uncontrolled intercurrent illness which in the opinion of the investigator poses
             unacceptably high risk when combined with study treatment, including but not limited
             to the following:

               -  Symptomatic congestive heart failure

               -  Unstable angina pectoris

               -  Severely impaired lung function

               -  Known history of active TB (Bacillus Tuberculosis)

               -  Uncontrolled diabetes as defined by fasting serum glucose >1.5 x ULN (NOTE:
                  Optimal glycemic control should be achieved before starting trial therapy.)

               -  Significant underlying liver disease such as severe cirrhosis or hepatic
                  impairment

               -  Has known psychiatric or substance abuse disorders that would interfere with
                  cooperation with the requirements of the trial.

          -  Has an active infection requiring systemic therapy prior to therapy initiation.

          -  Has a history or current evidence of any condition, therapy, or laboratory abnormality
             that might confound the results of the trial, interfere with the subject's
             participation for the full duration of the trial, or is not in the best interest of
             the subject to participate, in the opinion of the treating investigator.

          -  Is pregnant or breastfeeding, or expecting to conceive or father children within the
             projected duration of the trial, starting with the pre-screening or screening visit
             through 120 days after the last dose of trial treatment.

          -  Has known history of or any evidence of active, non-infectious pneumonitis.

          -  Currently uncontrolled hyper/hypothyroidism or hyper/hypocortism if in the opinion of
             the investigator they pose unacceptably high risk when combined with study treatment.

          -  Received live vaccine or live attenuated vaccine within 30 days prior to registration.
             NOTE: Administration of killed vaccines is allowed.

          -  Prior pancreatitis that was symptomatic or required medical intervention ≤ 6 months
             prior to registration (known toxicity of pembrolizumab).

          -  Prior enteritis that was symptomatic or required medical intervention ≤ 6 months prior
             to registration (known toxicity of pembrolizumab).

          -  Pre-existing motor or sensory neurotoxicity grade 3 or higher.

          -  For patients who received palliative RT, there must be no evidence of disease
             progression on clinical or imaging exams for at least two weeks prior to first dose of
             treatment.

          -  Parenteral or oral corticosteroids in the last two weeks prior to starting study
             treatment.

          -  Prior solid organ or allogeneic transplant.