OUTLINE: This is a multicenter study. Patients are stratified according to low vs high weight. Patients with severe aplastic anemia, myelodysplastic syndrome, or bone marrow failure receive cyclophosphamide IV over 1 hour on days -6 to -3 or melphalan IV over 20 minutes on days -4 to -2, antithymocyte globulin (ATG) IV over 4 hours or methylprednisolone IV over 1 hour twice a day on days -3 to -1, and total lymphoid irradiation on day -1. On day 0, patients receive umbilical cord blood (UCB) infusion. Patients with inborn errors of metabolism or inherited hematopoietic disorders receive oral busulfan every 6 hours on days -9 to -6, cyclophosphamide IV over 1 hour on days -5 to -2 or melphalan IV over 20 minutes on days -4 to -2, and ATG IV over 4 hours or methylprednisolone IV over 1 hour on days -3 to -1. On day 0, patients receive UCB infusion. Patients with Fanconi's anemia receive ATG IV over 4 hours or methylprednisolone IV over 1 hour on days -6 to -1, cyclophosphamide IV over 1 hour on days -5 to -2, thoracoabdominal irradiation on day -1, and then the UCB infusion on day 0. Patients also receive cyclosporine and methylprednisolone beginning on day -2 and continuing as necessary as graft-versus-host disease prophylaxis. Patients are followed indefinitely for survival and late toxicity.