This phase II trial studies how well giving an umbilical cord blood transplant together with cyclophosphamide, fludarabine phosphate, and total-body irradiation (TBI) works in treating patients with hematologic disease. Giving chemotherapy, such as cyclophosphamide and fludarabine phosphate, and TBI before a donor umbilical cord blood transplant helps stop the growth of cancer and abnormal cells and helps 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. Sometimes the transplanted cells from a donor can make an immune response against the body's normal cells. Giving cyclosporine and mycophenolate mofetil after transplant may stop this from happening.