This phase II trial investigates how well a naive T cell depleted graft work for the reduction of graft versus host disease in patients with non-malignant diseases requiring hematopoietic cell transplantation. Giving chemotherapy and total-body irradiation before a donor peripheral blood stem cell transplant helps stop the growth of cells in the bone marrow, including normal blood-forming cells (stem cells) and cancer cells. It may also stop the patient's immune system from rejecting the donor's stem cells. When the healthy stem cells from a donor are infused into the patient, they may help the patient's bone marrow make stem cells, red blood cells, white blood cells, and platelets. The donated stem cells may also replace the patient's immune cells and help destroy any remaining cancer cells.