Pembrolizumab, Capecitabine, and Radiation Therapy in Treating Patients With Mismatch-Repair Deficient and Epstein-Barr Virus Positive Gastric Cancer

Inclusion Criteria:

* Written informed consent for the trial
* Must have operable gastric adenocarcinoma, T2-T4a, N0-N3, M0
* Staging evaluation within 42 days of enrollment consisting of staging laparoscopy, computed tomography (CT) scan of the abdomen and pelvis, and positron emission tomography (PET) scan will be acquired; also endoscopic ultrasound (EUS) tumor and nodal staging will be required
* Mismatch repair deficiency as identified by immunohistochemistry or other institutional standard, or Epstein-Barr virus positivity as determined by in situ hybridization or other institutional standard
* Be willing to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion; newly-obtained is defined as a specimen obtained up to 10 weeks (70 days) prior to initiation of treatment on day 1; subjects for whom newly-obtained samples cannot be provided (e.g. inaccessible or subject safety concern) may submit an archived specimen (if available from prior biopsy) only upon agreement from the sponsor
* Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) performance scale
* Absolute neutrophil count (ANC) \>= 1,500 /mcL
* Platelets \>= 100,000 / mcL
* Hemoglobin \>= 7 g/dL or \>= 5.6 mmol/L with transfusion or erythropoietin (EPO) dependency
* Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance (glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN
* Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN
* Albumin \>= 2.5 mg/dL
* International normalized ratio (INR) or prothrombin time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants
* Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants
* Female subject of childbearing potential should have a negative urine or serum pregnancy within 7 days of enrollment; if the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* Female subjects of childbearing potential must be willing to use an adequate method of contraception for the course of the study through 120 days after the last dose of study medication; Note: abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject
* Male subjects of childbearing potential 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 usual lifestyle and preferred contraception for the subject

Exclusion Criteria:

* Presence of metastatic disease is not allowed; subjects must be evaluated with imaging consisting of CT scan and PET scan prior to enrollment for protocol therapy to exclude metastatic disease
* Prior chemotherapy, adjuvant therapy, or radiotherapy for gastric cancer
* Prior radiotherapy that overlaps with planned radiotherapy portal
* Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment
* Has a known history of active TB (Bacillus tuberculosis)
* Hypersensitivity to pembrolizumab or any of its excipients
* Has had a prior anti-cancer monoclonal antibody (mAb) within 4 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 4 weeks earlier
* Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study day 1 or who has not recovered (i.e., =\< grade 1 or at baseline) from adverse events due to a previously administered agent

  * Note: subjects with =\< grade 2 neuropathy are an exception to this criterion and may qualify for the study
  * Note: if subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy
* Has a known additional malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer
* 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
* Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis
* Evidence of interstitial lung disease
* Has an active infection requiring systemic therapy
* 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
* Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial
* 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 received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent
* Has a known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies)
* Has known active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus (HCV) RNA \[qualitative\] is detected)
* Has received a live vaccine within 30 days of planned start of study therapy; Note: seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist) are live attenuated vaccines, and are not allowed