This phase I trial studies the side effects and best dose of pembrolizumab when given together with chemoradiotherapy or radiation therapy in treating patients with small cell lung cancer. Immunotherapy with monoclonal antibodies, such as pembrolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Radiation therapy uses high energy x-rays to kill cancer cells and shrink tumors. Giving chemotherapy with radiation therapy may kill more cancer cells. Giving pembrolizumab with chemoradiotherapy or radiation therapy may be a better treatment for small cell lung cancer.