Inclusion Criteria:

          -  Histologically or cytologically confirmed adenocarcinoma involving the
             gastroesophageal junction or gastric cardia

          -  Central pathology review to determine evaluability of archived
             esophagogastroduodenoscopy (EGD)/biopsy sample

               -  NOTE: If archived sample was collected > 8 weeks prior to pre-registration (reg),
                  is not available in a timely manner, or was collected outside of Mayo Clinic and
                  considered unevaluable, then baseline EGD with primary tumor biopsy at Mayo
                  Clinic must be performed unless clinically contraindicated; patient is allowed to
                  enroll regardless of whether this Mayo Clinic tissue sample is evaluable; (Only 1
                  EGD with primary tumor biopsy performed at Mayo Clinic =< 8 weeks prior to
                  pre-reg is required)

               -  NOTE: For both archival or newly obtained tissue, only biopsies are adequate
                  (fine needle aspiration [FNA] is not adequate)

          -  Willing to provide mandatory tissue samples for research purposes

          -  Baseline imaging with an fludeoxyglucose (FDG)-positron emission tomography (PET) scan
             negative for distant metastatic disease must be obtained =< 28 days prior to
             registration

          -  Surgically resectable (T2N0, T3N0, Tany with node positivity, M0), as determined by
             endoscopic ultrasound (EUS) and the following minimum diagnostic work-up:

               -  Whole-body PET/computed tomography (CT) (PET/CT of skull base to mid-thigh is
                  acceptable)

               -  EUS =< 21 days prior to registration

               -  NOTE: Patients may have regional adenopathy including para-esophageal, gastric,
                  gastrohepatic and celiac nodes; if celiac adenopathy is present, it must be < 2
                  cm

               -  NOTE: If patient unable to have PET/CT then CT chest/abdomen/pelvis with contrast
                  (preferred) or MRI chest/abdomen/pelvis with contrast

          -  Surgical consultation at enrolling site to confirm that patient will be able to
             undergo curative resection after completion of chemoradiation =< 56 days prior to
             registration

               -  Tumor is amenable to standard resection and reconstruction

          -  Radiation oncology consultation at enrolling site to confirm that disease can be
             encompassed in a radiotherapy field =< 56 days prior to registration

               -  NOTE: Radiotherapy quality assurance rapid review must be performed before the
                  first fraction of radiation therapy (RT) is administered; if RT constraints
                  cannot be met, the patient will be removed from the protocol prior to treatment

          -  Consultation with a medical oncologist at enrolling site =< 56 days prior to
             registration, with determination that treatment with neoadjuvant chemoradiotherapy
             with weekly carboplatin and paclitaxel is considered acceptable

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1

          -  Adequate oral intake and nutritional status without current or likely need for enteral
             or parenteral feeding during chemoradiation or the preoperative period

          -  Pre-treatment pulmonary function tests (PFTs), collected =< 90 days prior to
             enrollment, must show forced expiratory volume in one second (FEV1) > 60% of predicted

          -  Adequate organ function =< 21 days prior to registration:

               -  Aspartate transaminase (AST) level =< 2.5 x upper limit of normal (ULN) and
                  alanine transaminase (ALT) =< 3 x upper limit of normal (ULN)

               -  Total bilirubin level of =< 1.5 x ULN

               -  Creatinine level =< 1.2 x ULN or creatinine clearance > 60 mL/min/1.73 m^2 for
                  patients with creatinine levels above or below the institutional normal

               -  Hemoglobin (Hgb) >= 9 g/dl without transfusion or epoetin dependency (=< 7 days
                  prior to assessment)

               -  Absolute neutrophil count >= 1.5 x 10^9/L

               -  Platelets >= 100 x 10^9/L

               -  Albumin >= 2.5 g/dl

          -  Female patients of childbearing potential must be willing to use 2 methods of birth
             control or be surgically sterile, or abstain from heterosexual activity for the course
             of the study through 120 days after the last dose of study medication

               -  NOTE: Patients of childbearing potential are those who have not been surgically
                  sterilized or have not been free from menses for > 1 year

          -  Female patients of childbearing potential must have a negative urine or serum
             pregnancy test =< 7 days prior to registration

               -  NOTE: If the urine test is positive or cannot be confirmed as negative, a serum
                  pregnancy test will be required

          -  Male patients must agree to use an adequate method of contraception starting with the
             first dose of study therapy through 120 days after the last dose of study therapy

               -  NOTE: Abstinence is acceptable if this is the established and preferred method of
                  contraception for the subject

          -  Provide signed written informed consent

          -  Willing to return to enrolling institution for follow-up

          -  Willing to provide mandatory tissue and blood samples for research purposes

               -  NOTE: Patients must be willing to provide at the time surgical resection; for
                  patients who do not undergo surgery, any on-study tumor biopsy obtained for
                  clinical purposes subsequent to the baseline biopsy must also be available for
                  analysis

        Exclusion Criteria:

          -  Tumor characteristics - any of the following are excluded:

               -  Evidence of distant metastases

               -  Tumors whose location is restricted to the tubular esophagus (i.e., without
                  involvement of the GEJ or cardia)

               -  Tumors whose proximal end are at the level of the carina or higher

               -  Invasion of the tracheobronchial tree or presence of tracheoesophageal fistula

               -  Palpable supraclavicular nodes, biopsy-proven involvement of supraclavicular
                  nodes, or radiographically involved supraclavicular nodes (> 1.5 cm in greatest
                  dimension)

               -  T1N0M0, T4Nany, or in situ carcinoma

               -  Tumor must not extend 5 or more cm into the stomach

          -  Received prior treatment or receiving current treatment for this malignancy

          -  Prior radiation to chest or abdomen, or to > 30% of the marrow cavity

          -  Inadequate caloric or fluid intake whereby there is a current or likely future need
             for enteral or parenteral feeding during chemoradiation or the preoperative period

          -  Major surgery =< 4 weeks prior to registration

          -  Active autoimmune disorders, including patients known to be human immunodeficiency
             virus (HIV) positive, or those requiring chronic steroid administration (excluding
             inhaled steroids)

          -  Uncontrolled diabetes (i.e., will interfere with the performance of the FDG PET/CT
             scans)

          -  Prior allergic reactions to drugs containing Cremophor, such as teniposide or
             cyclosporine, or study drugs involved in this protocol, or to a monoclonal antibody or
             prior hypersensitivity to platinum-containing agents

          -  Heart conditions - any of the following:

               -  Any atrial fibrillation =< 3 months prior to registration

               -  Unstable angina =< 12 months prior to registration

               -  Prior symptomatic congestive heart failure

               -  Documented myocardial infarction =< 6 months prior to registration (pretreatment
                  electrocardiogram [ECG] evidence of infarct only will not exclude patients)

               -  Prior significant ventricular arrhythmia requiring medication

               -  Prior 2nd or 3rd degree heart block or other types of clinically significant
                  conduction delay =< 6 months prior to registration

               -  Clinically significant pericardial disease (including pericardial effusion,
                  pericarditis) or cardiac valvular disease =< 12 months prior to registration

               -  NOTE: As part of history and physical, all patients must be assessed for signs or
                  symptoms of cardiac disease, or for prior history of cardiac disease; these
                  conditions include but are not limited to diseases related to cardiac valves,
                  pericardium, myocardium, atrioventricular delays or arrhythmias; it is strongly
                  recommended that signs or symptoms of potentially clinically significant disease
                  be evaluated with comprehensive cardiac echo

          -  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)

          -  Uncontrolled hyper/hypothyroidism or hyper/hypocorticism =< 6 months prior to
             registration (known toxicity of pembrolizumab)

          -  Pulmonary conditions - any of the following:

               -  Respiratory condition that required any oxygen supplementation =< 6 months prior
                  to registration

               -  Prior or current pneumonitis

               -  Clinically significant pulmonary hypertension =< 12 months prior to registration

               -  Lung infection requiring treatment =< 3 months prior to registration

               -  Pulmonary embolism requiring treatment =< 6 months prior to registration

               -  Pleural effusion requiring drainage =< 12 months prior to registration

          -  Prior fistula within thorax, including bronchoalveolar or esophageal

          -  Body mass index (BMI) >= 35 mg/m^2 =< 56 days prior to registration

          -  Pre-existing motor or sensory neurotoxicity greater than Common Terminology Criteria
             for Adverse Events (CTCAE) grade 1

          -  Acute bacterial, viral, or fungal infection requiring treatment at the time of
             registration

          -  Active infection or other serious underlying medical condition which would impair the
             ability of the patient to receive the planned treatment

          -  Uncontrolled intercurrent illness including, but not limited to, psychiatric
             illness/social situations, or other co-morbid systemic illnesses or severe concurrent
             diseases 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

          -  Prior malignancy =< 5 years prior to registration (except non-melanotic skin cancer or
             carcinoma-in-situ of the cervix) (must be disease free for a minimum of 5 years); if
             there is a history of prior malignancy, patient must not be receiving other specific
             treatment (other than hormonal therapy) for cancer

          -  Dementia or altered mental status that would prohibit the understanding and giving of
             informed consent

          -  Any of the following because this study involves an agent where the genotoxic,
             mutagenic and teratogenic effects are unknown:

               -  Pregnant or breastfeeding

               -  Patient of childbearing potential who is unwilling to employ adequate
                  contraception

          -  Received live vaccine =< 30 days prior to registration