Umbilical Cord Blood for Stem Cell Transplantation in Treating Young Patients With Malignant or Nonmalignant Diseases

OBJECTIVES: Primary * Determine the impact of the use of umbilical cord blood as a source of hematopoietic stem cells for children with life-threatening oncologic, hematologic, or genetic/metabolic disorders in need of a stem cell transplant. * Compare the incidence of graft-versus-host disease in patients receiving cord blood transplants in this study with historical data for unrelated donor stem cell transplants. * Compare the incidence of engraftment in patients receiving cord blood transplants in this study with historical data for unrelated donor stem cell transplants. OUTLINE: * Preparative therapy: Patients are treated on 1 of 4 preparative therapy regimens. * Regimen A: Patients undergo total body irradiation (TBI) two times daily on days -7 to -4. Patients receive cyclophosphamide IV over 30-60 minutes on days -3 and -2 and anti-thymocyte globulin (ATG) IV over at least 6 hours on days -3 to -1. * Regimen B (patients who do not receive TBI): Patients receive oral busulfan 4 times daily on days -8 to -5, and ATG IV over at least 6 hours and melphalan IV over 15-20 minutes on days -4 to -2. * Regimen C (patients with Fanconi's anemia and related disorders): Patients undergo TBI on day -6. Patients receive ATG IV over at least 6 hours and methylprednisolone IV on days -5 to -1 and fludarabine IV over 30 minutes and cyclophosphamide IV over 30-60 minutes on days -5 to -2. * Regimen D: Patients receive oral or IV busulfan 4 times daily on days -9 to -5, ATG IV over at least 6 hours on days -5 to -3, and cyclophosphamide IV over 30-60 minutes on days -5 to -2. * Cord blood transplant: All patients undergo umbilical cord blood transplantation on day 0. * Graft-versus-host disease prophylaxis: Patients receive oral or IV cyclosporine twice daily beginning on day -1. Patients also receive methylprednisolone IV twice daily beginning on day 5 and continuing until at least day 28. PROJECTED ACCRUAL: A total of 25 patients will be accrued for this study.