Abemaciclib in Treating Patients With Advanced, Refractory, and Unresectable Digestive System Neuroendocrine Tumors

Inclusion Criteria: * Histologically confirmed GEP NET, radiographically progressed on at least one line of standard therapy within the past 12 months * Primary tumors may be in: pancreas, foregut (esophagus, stomach, duodenum), midgut (small intestine, appendix), hindgut (large intestine, rectum), or unknown origin * Tumors may be functional (associated with clinical symptoms of hormone secretion) or non-functional * Well-differentiated NET with low grade (Ki67 index \< 3% or mitotic index \< 2 mitoses/10 high power field \[HPF\]), intermediate grade (Ki67 index 3-20% or mitotic index 2-20 mitoses/10 HPF), or high grade (Ki67 21% to ≤ 55% of mitotic index 21-55% mitoses/10 HPF). In cases where pathology reports call out only a "high grade neuroendocrine carcinoma", such patients are eligible only if well differentiated status is confirmed by a board-certified pathologist AND Ki-67 is ≤ 55% * Metastatic or locally advanced unresectable disease * Measurable disease by computed tomography (CT) or magnetic resonance imaging (MRI) as per Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1 * Prior or concurrent therapy with somatostatin analogs (SSAs) is allowed. If concurrent therapy, dose must be stable for at least 2 months * Patients with carcinoid syndrome must have symptoms controlled with stable doses of SSAs for at least 2 months \* Telotristat is not allowed * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2 * Able to swallow oral medications * Absolute neutrophil count \>= 1500/uL * Platelet count \>= 100,000/uL (without platelet transfusion for at least two weeks) * Hemoglobin \>= 8 g/dL (blood transfusion is not allowed the day before or on the day of study treatment) * Total bilirubin =\< 1.5 times upper limit of normal (ULN) * Transaminases (aspartate aminotransferase \[AST\] and/or alanine aminotransferase \[ALT\]) =\< 3 x upper limit of normal (ULN) (=\< 5 x ULN if liver metastases) * Patients with Gilbert's syndrome with a total bilirubin =\< 2.0 times ULN and direct bilirubin within normal limits are permitted * International normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.5 x ULN * Creatinine \> 30 mL/min * Ability to understand and sign the consent form * Women of child-bearing potential must: * Have a negative serum pregnancy test within 7 days prior to initiation of treatment, and * Agree to use a highly effective method of contraception during the study and for at least 3 weeks following the last dose of study drug * Men must be sterile or agree to use a highly effective method of contraception during the study and for at least 3 weeks following the last dose of study drug Exclusion Criteria: * Presence of poorly differentiated neuroendocrine carcinoma (NEC) or mixed adenoneuroendocrine carcinomas (MANECs) * Prior treatment with abemaciclib or other CDK4/6 inhibitors * Known hypersensitivity to abemaciclib or its components * Receipt of any therapy or investigational agent within 4 weeks prior to study registration, except SSAs * Any surgery, radiation, or embolization within 4 weeks * Peptide receptor radionuclide therapy (PRRT) within 6 weeks * Patients receiving other investigational agents * Patients who have not recovered from adverse events of prior therapy to =\< grade 1 (National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\] 5), except for alopecia or grade =\< 2 peripheral neuropathy prior to study treatment initiation. Subjects must have fully recovered from the acute effects of any prior radiotherapy * Patients with untreated or symptomatic brain metastases (must be off corticosteroids for \>= 4 weeks) * Uncontrolled or untreated intercurrent illness including, but not limited to, active bacterial or fungal infection, congestive heart failure, severe/unstable angina, syncope of cardiac etiology, ventricular arrythmia (including but not limited to ventricular tachycardia, ventricular fibrillation), history of cardiac arrest, interstitial lung disease, severe dyspnea at rest or requiring oxygen supplementation, arterial or venous thrombotic event, pre-existing chronic condition resulting in baseline grade \>= 2 diarrhea, or psychiatric illness/social situations that would limit compliance with study requirements * Gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for intravenous (IV) alimentation, prior surgical procedures involving stomach or small bowel in the last 28 days, active peptic ulcer disease, Crohn's disease or ulcerative colitis * Severe renal impairment (e.g. estimated creatinine clearance \< 30ml/min) * Known history of infection with human immunodeficiency virus (HIV) * Active untreated infection with hepatitis B virus (i.e. hepatitis B surface antigen positive) or hepatitis C virus (i.e. hepatitis C antibody and ribonucleic acid \[RNA\] positive) * Other malignancy diagnosed or recurrent in the past 3 years (except non-melanoma skin cancer and in-situ cervical cancer) * Pregnancy or breast-feeding