TAS 102 in Combination With Ramucirumab in Advanced, Refractory Gastric or Gastroesophageal Junction (GEJ) Adenocarcinoma

Inclusion Criteria: * Histologically confirmed diagnosis of gastric and GEJ adenocarcinoma. * Measurable or non-measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1. Stage IV or recurrent disease is required. An irradiated lesion is considered evaluable only if it has shown enlargement since the completion of last radiation. * Participants must have received and progressed with prior therapy. Prior therapy with ramucirumab is not allowed. Participants must have recovered from the toxic effects of the previous anti-cancer chemotherapy (with the exception of alopecia). * Eastern Cooperative Oncology Group (ECOG) Performance score 0 or 1 * Estimated life expectancy \> 3 months * Adequate bone marrow, liver and renal function as assessed by the following: Hemoglobin \> 8.0 g/dl, Absolute neutrophil count (ANC) \> 1,000/mm3 independent of growth factor support, Platelet count \> 100,000/mm3, Total bilirubin \< 1.5 times upper limits of normal (ULN) unless bilirubin rise is due to Gilbert's syndrome or of non-hepatic origin, aspartate transaminase (AST), alanine transaminase (ALT) and Alkaline Phosphatase ≤2.5 times the ULN ( ≤5 x ULN for patients with liver involvement), Creatinine clearance ≥ 30 ml/min. * Participants must not have had chemotherapy,major surgery, monoclonal antibody therapy or experimental therapy within the 28 days prior to the start of TAS 102 administration. * Women of childbearing potential must have a negative serum pregnancy test performed within 7 days prior to the start of study drug. Post-menopausal women (defined as no menses for at least 1 year) and surgically sterilized women are not required to undergo a pregnancy test. * Participants (men and women) of childbearing potential must agree to use adequate contraception beginning at the signing of the informed consent form until at least 4 months for both females and males after the last dose of study drug. The definition of adequate contraception will be based on the judgment of the principal investigator or a designated associate. * Participants must be able to understand and be willing to sign the written informed consent form. A signed informed consent form must be appropriately obtained prior to the conduct of any trial-specific procedure. Subjects must be willing and able to comply with scheduled visits, treatment schedule, laboratory testing, and other study requirements. Exclusion Criteria: * Participants with active Central Nervous System (CNS) metastases are excluded. If CNS metastases are treated and participants are at neurologic baseline for at least 2 weeks prior to enrollment, they will be eligible but will need a Brain MRI prior to enrollment. * Women who are pregnant or breast-feeding. * Prior therapy with ramucirumab, bevacizumab, regorafenib or TAS 102. * Previous or concurrent cancer within 3 years prior to treatment start EXCEPT for curatively treated cervical cancer in situ, non-melanoma skin cancer, superficial bladder 15 tumors \[Ta (non-invasive tumor), Tis (carcinoma in situ) and T1 (tumor invades lamina propria)\]. * Uncontrolled hypertension (systolic BP \>140 mm Hg or diastolic BP \>90 mm Hg on repeated measurement despite optimal medical management) * Any hemorrhage or bleeding event ≥ NCI CTCAE Grade 3 within 4 weeks prior to start of study medication. * Any hemorrhage or bleeding event ≥ NCI CTCAE Grade 3 within 4 weeks prior to start of study medication. * Persistent proteinuria ≥ Grade 3 on repeated measurement. * Substance abuse, medical, psychological or social conditions that may interfere with the patient's participation in the study or evaluation of the study results. * Participants with an arterial thrombotic or thromboembolic event within 12 months of informed consent. * Ascites, pleural effusion, or pericardial fluid requiring drainage in the last 4 weeks * Known history of human immunodeficiency virus (HIV) infection or current chronic or active hepatitis B or C infection requiring treatment with antiviral therapy. * Clinically significant cardiovascular disease such as unstable angina, uncontrolled or symptomatic arrhythmia, congestive heart failure, any Class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification, or history of myocardial infarction within 6 months prior to first dose with study drug. * Unable to swallow capsules or disease significantly affecting gastrointestinal function and/or inhibiting small intestine absorption such as; malabsorption syndrome, resection of the small bowel, or poorly controlled inflammatory bowel disease affecting the small intestine. * Child-Pugh B cirrhosis (or worse) or a history of hepatic encephalopathy * History of stroke or intracranial hemorrhage within 6 months prior to enrollment. * Any illness or medical conditions that are unstable or could jeopardize the safety of the patient and his/her compliance in the study. * Major surgery or a wound that has not fully healed within 4 weeks of enrollment. * Patients who need anticancer chemotherapy other than the study drugs during the study or within 4 weeks of study enrollment. Anti-cancer therapy is defined as any agent or combination of agents with clinically proven anti-tumor activity administered by any route with the purpose of affecting the malignancy, either directly or indirectly, including palliative and therapeutic endpoints. * Patients who need hormonal therapy during the study or within 2 weeks of first study enrollment. * Patients who need radiotherapy to target lesions during study or within 2 weeks of enrollment. * Patients who had bone marrow transplant or stem cell rescue. * Patients who need investigational drug therapy outside of this trial during or within 4 weeks of first study treatment.