Brief Summary: This phase Ib trial test effects of aldesleukin in combination with nivolumab and standard chemotherapy in treating patients with gastric cancer that has spread to the tissue lining of the abdomen (peritoneal metastasis). Aldesleukin is similar to a protein that naturally exists in the body that stimulates the immune system to fight infections. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Chemotherapy drugs, such as leucovorin calcium, fluorouracil, and oxaliplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving aldesleukin in combination with nivolumab and standard chemotherapy may work better in treating patients with gastric cancer with peritoneal metastasis.

Detailed Description: PRIMARY OBJECTIVE: I. To evaluate the reduction in the peritoneal carcinomatosis index (PCI) after completion of the study treatment. SECONDARY OBJECTIVES: I. Evaluate histological response of the peritoneal metastasis to study treatment using the peritoneal regression grading score (PRGS). II. Assess overall survival (OS). III. Assess progression-free survival (PFS). IV. Assess safety and tolerability of the study regimen. TERTIARY AND CORRELATIVE RESEARCH OBJECTIVES: I. Assess the number and percentage of patients that successfully undergo complete cytoreductive surgery after treatment. II. Evaluate for helper T cell, cytotoxic T cell, natural killer (NK) cells as well as T-reg cells in blood and peritoneal fluid. III. Evaluate the neutrophilic, lymphocytic, and eosinophilic infiltration of tumor using a standardized classification. OUTLINE: Patients receive aldesleukin intraperitoneally (IP) on study. Patients also receive standard of care nivolumab intravenously (IV), leucovorin calcium IV, fluorouracil IV, and oxaliplatin IV on study. Patients also undergo diagnostic laparoscopy with biopsy, positron emission tomography (PET)/computed tomography (CT) or PET/magnetic resonance imaging (MRI), and collection of blood and tissue samples throughout the trial.