Immunotherapy + Radiation in Resectable Soft Tissue Sarcoma

Inclusion Criteria: * Diagnosed with biopsy-proven soft-tissue sarcoma with confirmatory pathology review at University of Rochester Medical Center (URMC) * 18 years or older. * Able to provide informed consent. * Resectable disease per surgical evaluation. * Neoadjuvant/preoperative radiotherapy has been recommended * Intermediate to high grade sarcoma on biopsy, tumor \> 5 cm in size by imaging * Willing to have blood draws for flow cytometry and Serametrix analysis. * Willing to receive neoadjuvant radiation therapy and subsequent surgical resection. * Patients with known human immunodeficiency virus (HIV) are eligible if the lymphocytes \> 350 CD4+ cells and no detectable viral load. * Women of childbearing potential (defined as any woman, who 1) has not undergone tubal ligation, a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months or has had menses at any time in the preceding 24 consecutive months): * Must not be pregnant or nursing * Must have a negative pregnancy test done within 7 days prior to registration as well as within 72 hrs. prior to receiving first dose of study medication * Women of childbearing potential must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 5 months after the last dose of nivolumab and/or ipilimumab. * Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. * Sexually active males must use at least two other accepted and highly-effective methods of contraception and/or agree to abstain from sexual intercourse for at least 7 months after the last dose of nivolumab and/or ipilimumab * Eastern Cooperative Oncology Group (ECOG) performance status 0 - 2 Exclusion Criteria: * Gastrointestinal stromal tumor, dermatofibrosarcoma protuberans, rhabdomyosarcoma, Ewing's sarcoma, low-grade sarcomas, osteosarcoma subtypes * History of radiation to the affected area * Evidence of metastatic disease prior to treatment * Any history of prior therapy with ipilimumab or nivolumab, or any agent targeting PD-1, PD-L1 or CTLA-4. * History of any the following: * Active known or suspected autoimmune disease * Active autoimmune colitis * Autoimmune pan hypopituitarism * Autoimmune adrenal insufficiency * Known active hepatitis B or C * Known active pulmonary disease with hypoxia defined as: * Oxygen saturation \< 85% on room air or * Oxygen saturation \< 88% despite supplemental oxygen * No systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of registration. * Second active malignancy, not including localized basal cell skin cancer, squamous cell skin cancer, in situ cervical cancer, ductal or lobular carcinoma in situ of the breast; patients with a history of other malignancies are eligible if they have been continuously disease-free for \> 3 years prior to the time of registration. * Absolute neutrophil count (ANC) \<= 1,500/mm\^3 * Platelet count \<= 100,000/mm\^3 * AST (Aspartate Aminotransferase) or ALT (Alanine Aminotransferase) =\> 2 x upper limit of normal (ULN) * Thyroid stimulating hormone (TSH) outside of normal limits; supplementation is acceptable to achieve a TSH within normal limits; in patients with abnormal TSH if free T4 is normal and patient is clinically euthyroid, patient is eligible. * Life expectancy under 5 years.