This phase II trial studies how well ramucirumab and paclitaxel or the FOLFIRI regimen (leucovorin calcium, fluorouracil, and irinotecan hydrochloride) work in treating patients with small bowel cancers that have spread extensively to other anatomic sites (advanced) or are no longer responding to treatment (refractory). Ramucirumab is a monoclonal antibody that attaches to and inhibits a molecule called VEGFR-2. This may restrain new blood vessel formation therefore reducing nutrient supply to tumor which may interfere with tumor cell growth and expansion. Drugs used in chemotherapy, such as paclitaxel, leucovorin calcium, fluorouracil, and irinotecan hydrochloride work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving Ramucirumab plus paclitaxel or FOLFIRI, may be helpful in treating advanced or refractory small bowel cancers and may help patients live longer.