Abemaciclib and Pembrolizumab in Locally Advanced Unresectable or Metastatic Gastroesophageal Adenocarcinoma: Big Ten Cancer Research Consortium BTCRC-GI18-149

Inclusion Criteria: * Patients with histologically confirmed metastatic or locally advanced unresectable gastric, gastroesophageal junction or esophageal adenocarcinoma. * Be willing and able to provide written informed consent for the trial. * Age \>= 18 years at the time of consent. * Prior treatment with at least two lines of systemic therapy for advanced disease. Patients who have received neoadjuvant or adjuvant therapy or definitive chemoradiation and had recurrence during or within 6 months of completion of all treatments may count adjuvant therapy as one chemotherapy line. * Presence of measurable disease based on RECIST 1.1 as determined by local site investigator/radiology assessment. * ECOG PS 0-1. * Patients must have discontinued all previous treatments for cancer (including cytotoxic chemotherapy, molecularly targeted therapy, radiotherapy, and investigational therapy). * Patients who received chemotherapy must have recovered (CTCAE Grade ≤1) from the acute effects of chemotherapy except for residual alopecia or Grade 2 peripheral neuropathy. A washout period of at least 21 days is required between last systemic therapy dose and treatment initiation per protocol. * A washout period of at least 14 days is required between end of radiotherapy and treatment initiation. * The patient is able to swallow oral medications. * Demonstrate adequate organ function as defined in the table in the protocol. * Females of childbearing potential must have a negative serum pregnancy test within 7 days prior to registration. NOTE: Females are considered of child bearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months. * Females of childbearing potential must be willing to abstain from heterosexual activity or to use 2 forms of effective methods of contraception from the time of informed consent until 60 days after treatment discontinuation. The two contraception methods can be comprised of two barrier methods, or a barrier method plus a hormonal method. * Men who are not surgically sterile (vasectomy) must agree to use an acceptable method of contraception. Male subjects with female sexual partners who are pregnant, possibly pregnant, or who could become pregnant during the study must agree to use condoms from the first dose of study drug through at least 60 days after the last dose of study drug. Total abstinence for the same time period is an acceptable alternative. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study. * Provided written informed consent and HIPAA authorization for release of personal health information, approved by an Institutional Review Board (IRB). NOTE: HIPAA authorization may be included in the informed consent or obtained separately. Exclusion Criteria: * Active autoimmune disease that has required systemic treatment in past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * History of prior therapy with CDK4 or CDK6 inhibitors or prior immune checkpoint inhibitors. * Patients with known microsatellite instability will be excluded. * Diagnosis of immunodeficiency or is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. The use of physiologic doses of corticosteroids may be approved after consultation with the study PI. * Serious preexisting medical condition(s) that would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, history of major surgical resection involving the stomach or small bowel, or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea). * Symptomatic central nervous system metastasis. Screening of asymptomatic patients is not required for enrollment. * Personal history of any of the following conditions: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest. * Known additional malignancy that progressed or required active treatment within the last 2 years. Exceptions include curatively treated basal cell and squamous cell carcinoma of the skin, curatively resected in situ cervical and/or breast cancers, in situ or intramucosal pharyngeal cancer, and Gleason 6 prostate cancer with PSA \<10. * Patients who have received a live vaccine within 30 days of planned start of pembrolizumab. Note: The killed virus vaccines used for seasonal influenza vaccines for injection are allowed; however intranasal influenza vaccines (e.g., FluMist®) are live attenuated vaccines, and are not allowed. * History of (non-infectious) pneumonitis that required steroids or current pneumonitis. * Active infection requiring systemic therapy. * Patients who are pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the trial, starting with the screening visit through 60 days after the last dose of pembrolizumab or abemaciclib. (NOTE: breast milk cannot be stored for future use while the mother is being treated on study.)