Pralatrexate and Oxaliplatin in Treating Patients With Unresectable or Metastatic Esophageal, Stomach, or Gastroesophageal Junction Cancer

Inclusion Criteria: * Histologically confirmed carcinoma of the esophagus, stomach or gastro-esophageal junction that is metastatic, or locally advanced and inoperable for cure; histological sub-types permitted include adenocarcinoma, squamous-cell carcinoma, or undifferentiated carcinoma; small-cell carcinoma variant is not eligible * No previous systemic therapy for metastatic or recurrent disease; therapy (chemotherapy, radiotherapy, or both) administered in the neo-adjuvant, adjuvant, or definitive setting for previously localized disease is permitted, provided it was completed more than 6 months prior to enrollment; palliative radiotherapy is permitted provided it is completed \>= 3 weeks prior to study therapy initiation * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Life expectancy \>= 12 weeks * Hemoglobin \>= 9 g/dl * Absolute neutrophil count \>= 1500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Serum creatinine =\< institutional upper limit normal (ULN) * Bilirubin =\< 1.5 x ULN * Transaminases =\< 3 x ULN; for documented liver metastases, transaminases up to 5 x ULN is permitted * No evidence of \>= grade 2 peripheral neuropathy * Patients with reproductive potential must be willing to use an adequate contraceptive method (e.g., abstinence, intrauterine device, oral contraceptives, barrier device with spermicide or surgical sterilization) during treatment and for three months after completing treatment; a negative pregnancy test is required for women of child-bearing potential; nursing women are ineligible * Written, informed consent Exclusion Criteria: * Hypersensitivity to platinum compounds * Uncontrolled inter-current illness including but not limited to active infection, symptomatic congestive heart failure, unstable angina, uncontrolled cardiac arrhythmia, or psychiatric illness that would limit compliance with study requirements * Presence of brain metastases * Patients with third-space (pleural, peritoneal) fluid not controllable with usual drainage methods are not eligible * History of second primary malignancy within 3 years prior to enrollment, except for in-situ cervix carcinoma or non-melanoma skin cancer * Undergone an allogeneic stem cell transplant