Nivolumab, Ipilimumab and Chemoradiation in Treating Patients With Resectable Gastric Cancer

Inclusion Criteria: * Subjects must have signed and dated an Institutional Review Board (IRB)/Independent Ethics Committee (IEC) approved written informed consent form in accordance with regulatory and institutional guidelines. This must be obtained before the performance of any protocol-related procedures that are not part of normal subject care. Subjects must be willing and able to comply with scheduled visits, treatment schedule, laboratory tests and other requirements of the study. * All subjects must have localized/eligible for surgery gastric cancer (GC) or GEJ carcinoma type III, with negative peritoneal washing. Subjects must have histologically confirmed predominant adenocarcinoma. The documentation of GEJ involvement can include biopsy, endoscopy, or imaging. * Subject must be previously untreated with systemic treatment (including HER 2 inhibitors) given as primary therapy for advanced or metastatic disease. No prior neoadjuvant chemotherapy, immunotherapy, radiotherapy and/or chemoradiotherapy are permitted. * Eastern Cooperative Oncology Group (ECOG) performance status score of 0 or 1. * Absolute neutrophil count (ANC) \>= 1,500/mcL (within 28 days of treatment initiation). * Platelets \>=100,000/mcL (within 28 days of treatment initiation). * Hemoglobin \>= 9 g/dL or \> 5.6 mmol/L; if patient is not actively bleeding and has hemoglobin of \< 9g/dL, patient can receive blood transfusion to increase hemoglobin to \>= 9g/dL (within 28 days of treatment initiation). * 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 subjects with creatinine levels \> 1.5 x institutional ULN (measured via 24-hour urine collection) (within 28 days of treatment initiation). Creatinine clearance should be calculated per institutional standard. * Serum total bilirubin =\< 1.5 X ULN (1.5 mg/dL or 25.65 umol/L) OR direct bilirubin \< ULN for subjects with total bilirubin levels \< 1.5 X ULN. Except patients with Gilbert's disease (\< 3 X ULN) (within 28 days of treatment initiation). * Aspartate aminotransferase AST (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR \< 5 X ULN for subjects with liver metastases (within 28 days of treatment initiation). * Albumin \>= 3 mg/dL (within 28 days of treatment initiation). * 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 (within 28 days of treatment initiation). * Women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 24 hours prior to the start of study drug. Women must not be breastfeeding. * Prior to chemotherapy and immunotherapy, WOCBP must agree to follow instructions for method(s) of contraception for the duration of study treatment with nivolumab and 5 months after the last dose of study treatment (i.e. 30 days (duration of ovulatory cycle plus the time required for the investigational drug to undergo approximately five half-lives). * Males who are sexually active with WOCBP must agree to follow instructions for methods of contraception for the duration of study treatment with nivolumab and 7 months after the last dose of study treatment (i.e. 90 days (duration of sperm turnover) plus the time required for the investigational drug to undergo approximately five half-lives). In addition, male subjects must be willing to refrain from sperm donation during this time. Exclusion Criteria: * Participants with an active, known or suspected autoimmune disease. Participants with type I diabetes mellitus, hypothyroidism only requiring hormone replacement, skin disorders (such as vitiligo, psoriasis, or alopecia) not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger are permitted to enroll. * Participants with a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days of study treatment. Inhaled or topical steroids, and adrenal replacement steroid doses \> 10 mg daily prednisone equivalent, are permitted in the absence of active autoimmune disease. * Known history of positive test for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS). * NOTE: Testing for HIV must be performed at sites where mandated locally. * White blood cell (WBC) \< 2000/uL. * Any positive test result for hepatitis B virus or hepatitis C virus indicating presence of virus, e.g. hepatitis B surface antigen (HBsAg Australia antigen) positive, or hepatitis C antibody (anti-HCV) positive (except if hepatitis C virus- ribonucleic acid \[HCV-RNA\] negative). * History of allergy or hypersensitivity to study drug components. * Prior malignancy active within the previous 3 years except for locally curable cancers that have been apparently cured, such as basal or squamous cell skin cancer, superficial bladder cancer, or carcinoma in situ of the prostate, cervix, or breast. * Patients with serious or uncontrolled medical disorders. * Prior treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, or anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or checkpoint pathways.