Inclusion Criteria:

          -  Patients must have histologically or cytologically confirmed well-differentiated
             neuroendocrine tumor with positive gallium 68 dotatate scan (as determined by the site
             principal investigator [PI]) within 6 months before enrollment. Lesions on the gallium
             68 dotatate scan are considered positive if the maximum standard uptake value (SUVmax)
             is > 2 times the mean standard uptake value (SUVmean) of normal liver parenchyma.
             Patients must have the gallium 68 dotatate scans available to send for central review
             before enrollment

          -  Patients must have measurable progressive disease based on RECIST criteria, version
             1.1, evidenced with CT/magnetic resonance imaging (MRI) scans obtained within 12
             months before enrollment

          -  Failure of at least one prior systemic cancer treatment with somatostatin analogs

          -  No prior exposure to peptide receptor radionuclide therapy

          -  Age >= 18 years. Because no dosing or adverse event data are currently available on
             the use of M3814 (peposertib) in combination with lutetium Lu 177 dotatate in patients
             < 18 years of age, children are excluded from this study

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 2 (Karnofsky >= 60%)

          -  White blood cell count >= 2,000/mcL

          -  Absolute neutrophil count >= 1,000/mcL

          -  Platelets >= 75,000/mcL

          -  Hemoglobin >= 8.0 g/dL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN)

          -  Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase [SGPT]) =<
             3 x institutional ULN

          -  Alkaline phosphatase =< 3 x institutional ULN

          -  Glomerular filtration rate (GFR) >= 50 mL/min/1.73 m^2 (per the Cockcroft-Gault [C-G])

          -  Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
             therapy with undetectable viral load within 6 months are eligible for this trial

          -  For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral
             load must be undetectable on suppressive therapy, if indicated

          -  Patients with a history of hepatitis C virus (HCV) infection must have been treated
             and cured. For patients with HCV infection who are currently on treatment, they are
             eligible if they have an undetectable HCV viral load

          -  Patients with a prior or concurrent malignancy whose natural history or treatment does
             not have the potential to interfere with the safety or efficacy assessment of the
             investigational regimen are eligible for this trial

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, patients should be class 2B or better

          -  Patients must have recovered from adverse events of previously administered
             therapeutic agents to grade 2 or less toxicity according to Common Terminology
             Criteria for Adverse Events (CTCAE) 5.0

          -  Pregnancy Precaution: Men and women should avoid pregnancy for seven months after the
             date of their last treatment with lutetium Lu 177 dotatate. It is noteworthy that
             beta-human chorionic gonadotropin (HCG) may be secreted by a small percentage of
             neuroendocrine tumor (NET)s, such that, in addition to being a pregnancy marker, it
             also is a tumor marker. Consequently, NET female patients with positive beta-HCG (> 5
             mIU/mL) at baseline can be eligible to enter the study and receive treatment if
             pregnancy can be excluded by lack of expected doubling of beta-HCG and negative pelvic
             ultrasound. Normally, in pregnant subjects beta-HCG doubles every 2 days during the
             first 4 weeks of pregnancy and every 3.5 days by Weeks 6 to 7. Women of childbearing
             potential include any female who has experienced menarche and who has not undergone
             successful surgical sterilization (hysterectomy, bilateral tubal ligation, or
             bilateral ovariectomy) or is not postmenopausal (defined as amenorrhea > 12
             consecutive months, and for women on hormone replacement therapy, only with a
             documented plasma follicle-stimulating hormone [FSH] level > 35 mIU/mL). Even women
             who are using oral, implanted, or injected contraceptive hormones, an intrauterine
             device (IUD), or barrier methods (diaphragm, condoms, spermicidal) to prevent
             pregnancy, are practicing abstinence, or where the partner is sterile (e.g.,
             vasectomy) should be considered to be of childbearing potential. Postmenopausal women
             who have fertilized eggs implanted are also considered to be of childbearing
             potential. Acceptable methods of contraception may include total abstinence at the
             discretion of the Investigator in cases where the age, career, lifestyle, or sexual
             orientation of the patient ensures compliance. Periodic abstinence (e.g., calendar,
             ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable
             methods of contraception. Reliable contraception (hormonal or barrier method of birth
             control; abstinence) should be maintained throughout the study and for 7 months after
             study treatment discontinuation. All women of childbearing potential and male partners
             must use a double-barrier method of birth control or practice continuous abstinence
             from heterosexual contact throughout the study and for seven months after the end of
             the last treatment

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible.

        Exclusion Criteria:

          -  Patients who have had major surgical procedures in the 4 weeks prior to enrollment

          -  Patients are excluded if they received prior systemic peptide receptor radionuclide
             therapy (PRRT)-based therapies

          -  Patients with an inability to swallow oral medications or gastrointestinal disease
             limiting absorption of oral agents

          -  Patients who are receiving any other investigational agents

          -  Known brain metastases, unless these metastases have been treated and stabilized for
             at least 4 weeks, prior to enrollment in the study. Patients with a history of brain
             metastases must have either a head CT with contrast or brain magnetic resonance
             imaging (MRI) to document stable disease prior to enrollment in the study

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to M3814 (peposertib) or lutetium Lu 177 dotatate

          -  Prior external beam radiotherapy to more than 50% of bone marrow (whole body) will be
             excluded, as determined by a radiation medicine physicist who will calculate the
             volume of bone marrow exposure in prior radiotherapy portals divided by the volume of
             total bone marrow harboring tissues. This ratio must be less than 50 percent

          -  Patients who cannot discontinue concomitant medications or herbal supplements that are
             strong inhibitors or strong inducers of cytochrome P450 (CYP) isoenzymes CYP3A4/5,
             CYP2C9, and CYP2C19. Concomitant use of CYP3A4/5 substrates with a narrow therapeutic
             index are also excluded. Patients may confer with the study doctor to determine if
             alternative medications can be used. The following categories of medications and
             herbal supplements must be discontinued for at least the specified period of time
             before the patient can be treated:

               -  Strong inducers of CYP3A4/5, CYP2C9 and CYP2C19: >= 3 weeks prior to study
                  treatment

               -  Strong inhibitors of CYP3A4/5, CYP2C9 and CYP2C19: >= 1 week prior to study
                  treatment

               -  Substrates of CYP3A4/5 with a narrow therapeutic index: >= 1 day prior to study
                  treatment Patients receiving any medications or substances that are inhibitors or
                  inducers of CYP450 enzymes are ineligible. Because the lists of these agents are
                  constantly changing, it is important to regularly consult a frequently-updated
                  medical reference. As part of the enrollment/informed consent procedures, the
                  patient will be counseled on the risk of interactions with other agents, and what
                  to do if new medications need to be prescribed or if the patient is considering a
                  new over-the-counter medicine or herbal product

          -  Patients who cannot discontinue concomitant proton-pump inhibitors (PPIs). Patients
             may confer with the study doctor to determine if such medications can be discontinued.
             These must be discontinued >= 5 days prior to study treatment. Patients do not need to
             discontinue calcium carbonate

          -  Patients with ongoing active infection or treatment with a live attenuated vaccine
             within 4 weeks of dosing. In addition, a negative polymerase chain reaction (PCR) test
             for Covid-19 infection is highly recommended before entering the study, and a close
             symptom follow up within the study

          -  Patients with uncontrolled intercurrent illness

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Pregnant women are excluded from this study because M3814 (peposertib) is a DNA-PK
             inhibitor and lutetium Lu 177 dotatate is a peptide receptor radionuclide therapy with
             the potential for teratogenic or abortifacient effects. Because there is an unknown
             but potential risk for adverse events in nursing infants secondary to treatment of the
             mother with M3814 (peposertib) and lutetium Lu 177 dotatate, breastfeeding should be
             discontinued if the mother is treated with M3814 (peposertib) and lutetium Lu 177
             dotatate and for 2.5 months following the last lutetium Lu 177 dotatate treatment

          -  Discontinue long-acting somatostatin analogs (e.g., long-acting octreotide) for at
             least 4 weeks prior to initiating lutetium Lu 177 dotatate. Long-acting somatostatin
             analogs will be allowed to continue if patient has a history of carcinoid syndrome and
             requires long-acting somatostatin analogs for control of his/her functional syndrome