Brief Summary: RATIONALE: Everolimus may stop the growth of stomach or esophageal cancer by blocking blood flow to the tumor. Drugs used in chemotherapy, such as leucovorin calcium, fluorouracil, and oxaliplatin, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing, or by stopping them from spreading. Giving everolimus together with combination chemotherapy may kill more tumor cells. PURPOSE: This phase I trial is studying the side effects and best dose of everolimus when given together with combination chemotherapy in treating patients with metastatic stomach or esophageal cancer that has spread to other places in the body.

Detailed Description: OBJECTIVES: I. To determine the maximum tolerated dose of everolimus to use in combination with mFOLFOX6 \[oxaliplatin, leucovorin (leucovorin calcium), 5-FU (fluorouracil)\]. II. To better describe the toxicities associated with the combination of everolimus with mFOLFOX6. III. To assess response rate and progression-free survival in this patient population. IV. To assess overall survival in patients with metastatic gastric, esophageal and gastroesophageal junction (GEJ) adenocarcinoma treated with the combination of mFOLFOX6 + everolimus. OUTLINE: This is a dose-escalation study of everolimus. Patients receive fluorouracil intravenously (IV) continuously over 46 hours, leucovorin calcium IV over 2 hours, and oxaliplatin IV over 2 hours on day 1. Patients also receive oral everolimus once daily on days 1-28. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up periodically for 1 year.