This pilot clinical trial studies total-body irradiation followed by cyclosporine and mycophenolate mofetil in treating patients with severe combined immunodeficiency (SCID) undergoing donor bone marrow transplant. Giving total-body irradiation (TBI) before a donor bone marrow transplant using stem cells that closely match the patient's stem cells, helps stop the growth of abnormal cells. It may also stop the patient's immune system from rejecting the donor's stem cells. The donated stem cells may mix with the patient's immune cells and help destroy any remaining abnormal cells. Sometimes the transplanted cells from a donor can also make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil after the transplant may stop this from happening.