Pralatrexate and Docetaxel in Treating Patients With Stage IV Esophageal or Gastroesophageal Cancer Who Have Failed Platinum-Based Therapy

Inclusion * Pathologically confirmed unresectable advanced or metastatic carcinoma of the esophagus or gastroesophageal junction * Established histological confirmation of squamous cell carcinoma or adenocarcinoma of the esophagus or gastroesophageal junction * Stage IV disease * Must have received platinum-based therapy; this includes definitive, adjuvant and metastatic treatments * No more than 3 chemotherapeutic treatment regimens permitted; this includes concurrent chemoradiation * Radiation therapy allowed if \> 4 weeks have elapsed * Must be off therapy for 4 weeks prior to enrollment * Measurable disease as defined by RECIST v 1.1 criteria * ECOG (Eastern Cooperative Oncology Group)PS(Performance status)of 0 to 2 * Predicted life expectancy of at least 12 weeks * Patients with reproductive potential must use an effective method to avoid pregnancy for the duration of the trial and for three months after completion of treatment * Marrow: ANC(absolute neutrophil count)\> 1,000/mm\^3 * Marrow: Hemoglobin \> 9.0 g/dl * Marrow: Platelet Count \> 100,000/mm\^3 * Renal: Serum creatinine =\< 1.5 g/dL * Hepatic: Serum bilirubin \< 1.5 x ULN(upper limit of normal) and AST (aspartate aminotransferase) and ALT (Alanine aminotransferase)=\< 2.5 x ULN * Prior minor surgeries (such as laparoscopies) must have occurred at least 14 days prior to study enrollment; prior minor procedures such as biopsies and mediport placement must have occurred at least 48 hours prior to study enrollment * All patients must have signed an informed consent indicating that they are aware of the neoplastic nature of their disease and have been informed of the procedures of the protocol, the experimental nature of the therapy, alternatives, potential benefits, side effects, risks, and discomforts * History of allergic reactions attributed to compounds of similar chemical composition to agents used in the study Exclusion * Pregnant or lactating women * Patients with any severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for study entry * Any malignant condition for which one has received treatment in the last two years excluding squamous or basal cell carcinomas * Patients with untreated brain metastases * Patients must not have grade 2 or higher baseline peripheral neuropathy, according to CTCAE v 4.0 * Patients must have NO continuing acute toxic effects (except alopecia) of any prior radiotherapy, chemotherapy, or surgical procedures; all such effects must have resolved to Common Terminology Criteria for Adverse Events (CTCAE v 4.0) Grade =\< 1 prior to study enrollment