OUTLINE: Patients undergo autologous bone marrow harvesting or peripheral stem cell collection prior to transplant regimen, unless the patient has acute leukemia in relapse, aplastic anemia, or myelodysplastic syndrome. Arm I: Patients eligible to undergo total body irradiation (TBI) first receive cyclophosphamide IV over 2 hours on days -5 and -4, then undergo TBI twice a day on days -3 to -1. Patients also receive antithymocyte globulin (ATG) IV over 10 hours on days -3 to -1. Cord blood is infused on day 0. Arm II: Patients not eligible to receive TBI receive oral busulfan every 6 hours on days -7 to -4 for a total of 16 doses. Cyclophosphamide, ATG, and cord blood are then administered as in arm I. All patients receive cyclosporine on days -2 to 180, methylprednisolone on days 5-180, and filgrastim (G-CSF) from day 1. Patients are followed weekly until day 180 and then monthly for 2 years.