MM-398 and Ramucirumab in Treating Patients With Gastric Cancer or Gastroesophageal Junction Adenocarcinoma

Inclusion Criteria: * The patient has a histopathologically or cytologically confirmed diagnosis of gastric or gastroesophageal junction (GEJ) adenocarcinoma * The patient has metastatic disease or locally advanced and unresectable disease that is evaluable, by radiological imaging per Response Evaluation Criteria in Solid Tumors, Version 1.1 (RECIST 1.1). (MRI candidates must have measurable disease in liver. * The patient has documented disease progression or intolerance of chemotherapy during first-line platinum-based chemotherapy for metastatic disease, or during or within 6 months after the last dose of neoadjuvant or adjuvant therapy * Additional lines of therapy are permitted as long as patient had received a platinum and/or a fluoropyrimidine component. Exposure to antiangiogenic agent (either approved or experimental treatment) is permitted. Exposure to antineoplastic therapy in addition to platinums and/or fluoropyrimidines is acceptable if the agents were used in the first-line metastatic or neoadjuvant/adjuvant setting * The patient has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count \>= 1,000/microliter (mcL) * Platelets \>= 75,000/mcL * Total bilirubin \< 1.5 x institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SPGT\]) =\< 2.5 x institutional upper limit of normal for patients without liver metastasis and =\< 5 x institutional upper limit of normal for patients without liver metastasis * Creatinine \< 1.5 x institutional upper limit of normal * International normalized ratio (INR) \< 1.5 x ULN and partial thromboplastin time \< 5 seconds above upper liming of normal (ULN), unless the patient is receiving anticoagulation therapy. Patients on full-dose anticoagulation therapy must be on a stable dose of oral anticoagulation therapy or low molecular weight heparin for a minimum of 14 days. Patients receiving warfarin must have an INR \< 3.0 x ULN and have no bleeding within 14 days prior to the first dose of ramucirumab or pathological condition that carries a high risk of bleeding, such as tumors involving major vessels or known varices * The patient has provided written informed consent prior to any study-specific procedures and is amenable to compliance with protocol schedules and testing * The patient has an estimated life expectancy of \> 12 weeks in the judgment of the investigator * The patient has resolution to grade 1 by Common Terminology Criteria for Adverse Events (CTCAE) Version 4.0 (National Cancer Institute \[NCI\] 2009), of all clinically significant toxic effects of previous anticancer therapy. Stable grade 2 neuropathy is permitted. Patients with nonserious and non-life threatening toxicities, such as alopecia, altered taste, or nail changes, can be considered * The patient, if male, is sterile (including vasectomy confirmed by post-vasectomy semen analysis) or agrees to use a reliable method of birth control and to not donate sperm during the study and for at least 12 weeks following the last dose of study treatment * The patient, if female, is surgically sterile, is postmenopausal, or agrees to use a highly effective method of birth control during the study and for 12 weeks following the last dose of study treatment. A "highly effective method of birth control" is defined as one that results in a low failure rate (that is, \< 1% per year) when used consistently and correctly * The patient, if female and of child-bearing potential, must have a negative serum or urine pregnancy test within 7 days prior to enrollment. A female of child-bearing potential is any woman (regardless of sexual orientation, having undergone a tubal ligation, or remaining celibate by choice) who meets the following criteria: * Has not undergone a hysterectomy or bilateral oophorectomy; or * Has not been naturally postmenopausal for at least 12 consecutive months (i.e., has had menses at any time in the preceding 12 consecutive months) Exclusion Criteria: * The patient has squamous cell or undifferentiated gastric cancer * The patient received systemic therapy within 28 days prior to enrollment * The patient received radiotherapy within 14 days prior to enrollment. Palliative radiotherapy during the study, if clinically indicated, can be considered after consultation with the principal investigator (PI). Any lesion requiring palliative radiotherapy or which has been previously irradiated cannot be considered for response assessment * The patient has documented brain metastases, leptomeningeal disease, or uncontrolled spinal cord compression. Screening of asymptomatic patients is not required. * The patient has a significant bleeding disorder or vasculitis or had a grade \>= 3 bleeding episode within 12 weeks prior to enrollment * The patient experienced any arterial thromboembolic event, including myocardial infarction, unstable angina, cerebrovascular accident, or transient ischemic attack, within 6 months prior to enrollment. * The patient has symptomatic congestive heart failure (CHF; New York Heart Association IIIV) or symptomatic or poorly controlled cardiac arrhythmia * The patient has uncontrolled hypertension, as defined in CTCAE version 4.0, prior to initiating study treatment, despite antihypertensive intervention. CTCAE Version 4.0 defines uncontrolled hypertension as grade \> 2 hypertension; clinically, the patient continues to experience elevated blood pressure (systolic \> 160 mmHg and/or diastolic \> 100 mmHg) despite medications) * The patient underwent major surgery within 28 days prior to initiation or central venous access device placement within 7 days prior to enrollment * The patient plans to undergo elective major surgery during the course of the trial * The patient has a history of gastrointestinal (GI) perforation or fistula within 6 months prior to enrollment * The patient has a history of inflammatory bowel disease or Crohn's disease requiring medical intervention (immunomodulatory or immunosuppressive medications or surgery) \< 12 months prior to enrollment * The patient has an acute or subacute bowel obstruction or history of chronic diarrhea that is considered clinically significant in the opinion of the investigator * The patient has either of the following: * Cirrhosis at a level of Child-Pugh B (or worse) * Cirrhosis (any degree) and a history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis. Clinically meaningful ascites is defined as ascites resulting from cirrhosis and requiring ongoing treatment with diuretics and/or paracentesis * The patient has a known allergy or hypersensitivity to any components of study treatment. * The patient is currently enrolled in a clinical trial involving an investigational product or unapproved use of a drug or is concurrently enrolled in any other type of medical research judged not to be scientifically or medically compatible with this study. Patients participating in surveys or observational studies are eligible to participate in this study * The patient has a serious illness or medical condition including, but not limited to, the following: * Known human immunodeficiency virus infection or acquired immunodeficiency syndrome-related illness * Active or uncontrolled clinically serious infection * The patient is pregnant or breastfeeding * The patient has a concurrent active malignancy other than the following: * Adequately treated non-melanomatous skin cancer * Curatively treated in situ carcinoma of the cervix or other noninvasive carcinoma or in situ neoplasm * The patient has a history of prior malignancy but has been disease free for \< 2 years prior to enrollment * The patient has a serious nonhealing: (a) wound, (b) peptic ulcer, or (c) bone fracture, within 28 days prior to enrollment * The patient experienced any grade 3 or 4 venous thromboembolic event (VTE) that is considered by the investigator to be life-threatening or that is symptomatic and not adequately treated by anticoagulation therapy, within 6 months prior to enrollment * Unable to undergo MRI due to presence of errant metal, cardiac pacemakers, pain pumps or other MRI incompatible devices (MRI group only) * Treated with parenteral iron in the previous 4 weeks (MRI group only) * Evidence of Iron overload as determined by (MRI group only) * Fasting transferrin saturation of \> 45% and/or * Serum ferritin levels \> 1000 ng/ml * The patient has any condition (for example, psychological, geographical, or medical) that does not permit compliance with the study and follow-up procedures or suggests that the patient is, in the investigator's opinion, not an appropriate candidate for the study