This study is for patients with non-resectable, recurrent, or metastatic well or moderately differentiated gastroenteropancreatic neuroendocrine tumors (GEP-NETs). The study will be conducted in two stages: 1) Safety Run-In and 2) Expanded Cohort. 1. Safety run-in: The first stage will include a safety run-in of 6 patients treated with pembrolizumab 200 mg intravenous (IV) every 3 weeks and lanreotide depot 90mg subcutaneous (SQ) every 3 weeks. Up to 6 patients at the Duke Cancer Institute will be accrued at the starting dose level. If one or less subject meets treatment-related discontinuation criteria (as specified in the protocol) during Cycle 1, then the study will proceed to the second stage, Expanded Cohort. 2. Expanded Cohort: Patients will be treated with pembrolizumab 200mg IV every 3 weeks and lanreotide depot 90mg SQ every 3 weeks as determined by the Safety Run-In Cohort.