Ramucirumab Plus Irinotecan for Previously Treated Advanced Gastric or Gastro-esophageal Junction Adenocarcinoma

Inclusion Criteria: * Histopathologically or cytologically confirmed diagnosis of gastric or gastroesophageal junction (GEJ) adenocarcinoma that is metastatic or locally advanced and unresectable. * Measurable disease defined as lesions that can be accurately measured in at least one dimension (longest diameter to be recorded) as ≥ 10 mm with CT scan (or MRI at the discretion of the principal investigator (PI)), as ≥ 20 mm by chest x-ray, or ≥ 10 mm with calipers by clinical exam. * Either primary or non-osseous metastatic site amenable for research biopsy for patients enrolled at Washington University, if safe and feasible, as confirmed by scheduling of biopsy procedure. Other methods to obtain appropriate cancer cells such as large-volume paracentesis or thoracentesis can be allowed at PI discretion. Biopsy or other procedures should be performed at least 7 days prior to C1D1. * Experienced documented objective radiographic or clinical disease progression during first-line therapy or within 4 months after the last dose of first-line therapy with any platinum/fluoropyrimidine doublet with or without anthracycline (epirubicin or doxorubicin) or taxane (docetaxel) for unresectable or metastatic disease. NOTE: This is not intended to be an exclusive list of allowed agents. The targeted therapies such as Herceptin and ADC, or immunotherapies without cytotoxic chemotherapy, are permitted. * At least 18 years of age. * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 1 * Normal bone marrow and organ function as defined below: * Absolute neutrophil count (ANC) ≥ 1,500/µL * Hemoglobin ≥ 9.0 g/dL (5.58 mmol/L) * Platelets ≥ 100,000/µL * Total bilirubin ≤ 1.5 mg/dL (25.65 µmol/L) * AST(SGOT)/ALT(SGPT) ≤ 3.0 x institutional upper limit of normal (IULN) (or ≤ 5.0 x IULN in the setting of liver metastases) * Creatinine ≤ 1.5 x IULN OR creatinine clearance ≥ 40 mL/min/1.73 m2 for patients with creatinine levels \> 1.5 x IULN (that is, if serum creatinine is \> 1.5 x IULN, a 24-hour urine collection to calculate creatinine clearance must be performed) * Urinary protein ≤ 1+ on dipstick or routine urinalysis (UA); if dipstick or routine UA is ≥ 2+, a 24-hour urine collection for protein must demonstrate \< 1000 mg of protein in 24 hours * Adequate coagulation function as defined by INR ≤ 1.5 and PTT ≤ 5 seconds above the ULN (unless receiving anticoagulation therapy). Patients receiving warfarin must be switched to low molecular weight heparin and have achieved stable coagulation profile prior to first dose of protocol therapy. * All clinically significant toxic effects (except peripheral neuropathy) of prior locoregional therapy, surgery, or other anticancer therapy have resolved to ≤ Common Terminology Criteria for Adverse Events (CTCAE) grade 1. * Women of childbearing potential and men must agree to use two forms of adequate contraception (hormonal or barrier method of birth control, abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while participating in this study, she must inform her treating physician immediately. Women of childbearing potential must have a negative serum pregnancy test within 7 days of study entry. * Ability to understand and willingness to sign an IRB approved written informed consent document (or that of legally authorized representative, if applicable). Exclusion Criteria: * Squamous cell or undifferentiated gastric cancer. * Received any chemotherapy (including irinotecan) other than platinum and fluoropyrimidine with or without anthracycline or taxane for advanced gastric or GEJ adenocarcinoma. * Received previous systemic chemotherapy with a cumulative dose of \> 900 mg/m\^2 of epirubicin or \> 400 mg/m\^2 of doxorubicin. * Received any previously systemic therapy (including investigational agents) targeting VEGF or the VEGFR signaling pathways. Other previous targeted therapies are permitted if stopped at least 28 days prior to start of treatment. * A history of other malignancy ≤ 3 years previous with the exception of basal cell or squamous cell carcinoma of the skin which were treated with local resection only or carcinoma in situ of the cervix or other solid tumors treated curatively and without evidence of recurrence. * Currently receiving any other investigational agents. * History or evidence of known brain metastases or carcinomatous meningitis. Patients with known brain metastases must be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * A history of allergic reactions attributed to compounds of similar chemical or biologic composition to monoclonal antibody treatment, any components used in the ramucirumab DP preparation, irinotecan, or other agents used in the study. * Any grade 3-4 GI bleeding within 3 months prior to enrollment. * History of gastrointestinal perforation and/or fistulae within 6 months prior to enrollment. * History of deep vein thrombosis, pulmonary embolism, or any other significant thromboembolism (venous port of catheter thrombosis or superficial venous thrombosis are not considered "significant") during the 3 months prior to enrollment. * History of any arterial thromboembolic event, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina within 6 months prior to enrollment. * Diagnosis of symptomatic congestive heart failure (NYHA II-IV) or symptomatic or poorly controlled cardiac arrhythmia. * Uncontrolled or poorly controlled hypertension (\> 160 mmHg systolic or \> 100 mmHg diastolic for \> 4 weeks) despite standard medical management. * Presence of serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to enrollment. * Major surgery within 28 days prior to first dose of protocol therapy. * Minor surgery/subcutaneous venous access device placement within 7 days prior to first dose of protocol therapy. * Receiving chronic antiplatelet therapy, including aspirin, nonsteroidal anti-inflammatory drugs (NSAIDs, including ibuprofen, naproxen, and others), dipyridamole or clopidogrel, or similar agents. Once-daily aspirin use (maximum dose 325 mg/day) is permitted. * The patient has elective or planned major surgery to be performed during the course of the clinical trial. * Bowel obstruction, history or presence of inflammatory enteropathy or extensive intestinal resection (hemicolectomy or extensive small intestine resection with chronic diarrhea), Crohn's disease, ulcerative colitis, or chronic diarrhea. * Cirrhosis at a level of Child-Pugh B (or worse) or cirrhosis (any degree) and a history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis (i.e. ascites from cirrhosis requiring diuretics or paracentesis). Patients with ascites not related to cirrhosis, such as malignant ascites, are allowed. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, metabolic disorders or other nonmalignant organ or systemic disease or secondary effects of cancer that induce a high medical risk and make assessment of survival uncertain, or psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant and/or breastfeeding. * Known HIV-positivity on combination antiretroviral therapy because of the potential for pharmacokinetic interactions with ramucirumab and irinotecan. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated.