Donor Umbilical Cord Blood Transplant in Treating Patients With Hematologic Cancer

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following: * Leukemia including, but not limited to, the following subtypes: * Chronic myelogenous leukemia * Acute myeloid leukemia (primary or secondary) * Acute lymphoblastic leukemia * Lymphoma * Myelodysplastic syndrome * Aplastic anemia * Fanconi's anemia * Diamond-Blackfan anemia * Inborn errors of metabolism (e.g., Hurler syndrome, Maroteaux-Lamy syndrome, myelodysplastic syndrome VI, metachromatic leukodystrophy, adrenoleukodystrophy, and globoid cell leukodystrophy) * Immune deficiency disorders * Patients must meet the eligibility requirements outlined in currently active treatment protocols of the University of Minnesota Bone Marrow Transplant Program * HLA-identical or 1, 2, or 3 antigen mismatched umbilical cord blood (UCB) donor with at least one DRB1 match available * Unrelated or related donor * UCB specimen must contain ≥ 2.0 x 10\^7 nucleated cells/kg patient body weight PATIENT CHARACTERISTICS: * See Disease Characteristics * No active infection * No history of HIV infection * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception PRIOR CONCURRENT THERAPY: * Not specified