Everolimus and Combination Chemotherapy in Treating Patients With Metastatic Stomach or Esophageal Cancer

Inclusion Criteria: * Patients must have histologically confirmed diagnosis of gastric, esophageal, and GEJ adenocarcinoma * Patients must have metastatic disease * Patients must not have received any chemotherapy for metastatic disease * Patients may have received prior adjuvant chemotherapy; completion of chemotherapy must be greater than 6 months from date of recurrent disease * Patients must have computed tomography (CT) or magnetic resonance imaging (MRI) scan; patients must have at least one measurable site of disease according to Response Evaluation Criteria in Solid Tumors (RECIST) criteria that has not been previously irradiated; if the patient has had previous radiation to the marker lesion(s), there must be evidence of progression since the radiation * Patients must have a ECOG Performance Status of 0-1 * Absolute neutrophil count (ANC) \> 1,500/mcl * Platelet count \> 100,000/mcl * Hemoglobin (Hg) \> 9 g/dL * Serum creatinine \< 1.5 mg/dl and/or Creatinine clearance \> 60 cc/min * Bilirubin \< 1.5 mg/dl * Serum glutamic oxaloacetic transaminase (SGOT) and/or serum glutamic pyruvic transaminase (SGPT) =\< 2.5 x institutional upper limit of normal (IULN) (=\< 5 x upper limit of normal \[ULN\] in patients with liver metastases) * Fasting serum cholesterol =\< 300 mg/dL or =\< 7.75 mmol/L * Fasting triglycerides =\< 2.5 x ULN; NOTE: in case one or both of these thresholds are exceeded, the patient can only be included after initiation of appropriate lipid lowering medication * Patients should have controlled diabetes as evidenced by hemoglobin (Hb)A1C =\< 8% * International normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.5; (anticoagulation is allowed if target INR =\< 1.5 on a stable dose of warfarin or on a stable dose of low molecular weight \[LMW\] heparin for \> 2 weeks at time of randomization) * Due to the possibility of harm to a fetus or nursing infant from this treatment regimen, patients must not be pregnant or nursing (or plan to become pregnant); women and men of reproductive potential must have agreed to use a highly effective contraceptive method for at least 8 weeks after treatment * Patients with risk factors for contraction hepatitis B or C should undergo screening prior to treatment on protocol. Patients with detectable viral titers are required to receive treatment for 4 weeks prior to starting protocol therapy. * Patients must be able to swallow pills * No other prior malignancy within the past 3 years is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated Stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years * All patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * Patients must have pathology specimen available for submission Exclusion Criteria: * Patients currently receiving anticancer therapies or who have received anticancer therapies within 4 weeks of the start of study drug (including chemotherapy, radiation therapy, antibody based therapy, etc) * Patients who have received prior treatment with an mammalian target of rapamycin (mTOR) inhibitor (sirolimus, temsirolimus, everolimus) * Patients with a known hypersensitivity to everolimus or other rapamycins (sirolimus, temsirolimus) or to its excipients * Patients, who have had a major surgery or significant traumatic injury within 4 weeks of start of study drug, patients who have not recovered from the side effects of any major surgery (defined as requiring general anesthesia) or patients that may require major surgery during the course of the study * Prior treatment with any investigational drug within the preceding 4 weeks * Patients receiving chronic, systemic treatment with corticosteroids (prednisone \> 10 mg per day) or another immunosuppressive agent; topical or inhaled corticosteroids are allowed * Patients should not receive immunization with attenuated live vaccines within one week of study entry or during study period * Uncontrolled brain or leptomeningeal metastases, including patients who continue to require glucocorticoids for brain or leptomeningeal metastases * Patients who have any severe and/or uncontrolled medical conditions or other conditions that could affect their participation in the study such as: * Symptomatic congestive heart failure of New York heart Association Class III or IV * Unstable angina pectoris, symptomatic congestive heart failure, myocardial infarction within 6 months of start of study drug, serious uncontrolled cardiac arrhythmia or any other clinically significant cardiac disease * Severely impaired lung function as defined as spirometry and diffusion lung capacity of carbon monoxide (DLCO) that is 50% of the normal predicted value and/or 02 saturation that is 88% or less at rest on room air * Uncontrolled diabetes as defined by fasting serum glucose \> 1.5 x ULN * Active (acute or chronic) or uncontrolled severe infections * Liver disease such as cirrhosis, chronic active hepatitis or chronic persistent hepatitis * A known history of human immunodeficiency virus (HIV) seropositivity * Impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of everolimus (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection) * Patients with an active, bleeding diathesis; history of noncompliance to medical regimens * Patients unwilling to or unable to comply with the protocol