Combination Chemotherapy Followed By Umbilical Cord Blood Transplantation in Treating Patients With Hematologic Cancer or Severe Aplastic Anemia

DISEASE CHARACTERISTICS: * One of the following histologically confirmed diagnoses: * Acquired severe aplastic anemia * Meets at least 2 of the following criteria: * Granulocyte count less than 500/mm\^3 * Platelet count less than 20,000/mm\^3 * Absolute reticulocyte count less than 20,000/mm\^3 (after correction for hematocrit) * Unresponsive to OR recurrent disease after prior treatment with anti-thymocyte globulin and/or cyclosporine * Acute myeloid leukemia (AML), meeting 1 of the following criteria: * Failed induction therapy * In first complete remission (CR) with any of the following high-risk features: * Stem cell or biphenotype classification (M0) * Erythroleukemia (M6) * Acute megakaryocytic leukemia (M7) * Cytogenetic markers indicative of poor prognosis * t(15;17) translocation and failed first-line induction therapy OR there is molecular evidence of persistent disease * t(8;21) and inv(16) translocations and failed first-line induction therapy * In early relapse\* * In second or subsequent remission * Recurrent disease after prior autologous stem cell transplantation (SCT) NOTE: \*No refractory relapse * Acute lymphoblastic leukemia, meeting 1 of the following criteria: * In early relapse\* * In second or subsequent remission * In first CR with the following high-risk features: * t(4;11) or t(9;22) translocation * Hyperleukocytosis (initial WBC …