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 greater than 30,000/mm^3) Failed to achieve CR by day 28 of standard induction therapy Recurrent disease after prior autologous SCT NOTE: *No refractory relapse Chronic myelogenous leukemia Chronic or accelerated phase that has failed medical management Blastic phase allowed after reinduction chemotherapy induces chronic phase Myelodysplastic syndromes meeting 1 of the following criteria: Refractory to medical management Presence of cytogenetic abnormalities predictive of transformation to acute leukemia, including the following: = 5q- = 7q- Monosomy 7 and trisomy 8 Evidence of evolution to AML (e.g., refractory anemia with excess blasts [RAEB], or RAEB in transformation) Chronic lymphocytic leukemia Refractory to treatment including fludarabine-based therapy Recurrent disease after prior autologous SCT Multiple myeloma Recurrent disease after prior autologous SCT Beyond first CR or failed induction therapy Disease is sensitive to pretransplantation cytoreduction Hodgkin's lymphoma Beyond first CR or failed induction therapy Disease is sensitive to pretransplantation cytoreduction Non-Hodgkin's lymphoma (NHL) Recurrent disease after prior autologous SCT Beyond first CR or failed induction therapy Disease is sensitive to pretransplantation cytoreduction Mantle zone NHL allowed after induction therapy Myeloproliferative disorders Refractory to medical management Allografting required unless grade 3 or greater myelofibrosis by bone marrow biopsy No HLA-matched sibling donor available Ineligible for a myeloablative conditioning regimen due to advanced age (over 55), extensive prior therapy, and/or other comorbidities If under age 55, must meet at least 1 of the following criteria: Received extensive prior therapy Organ toxicity or infection precluding eligibility for allogeneic transplantation with full ablation conditioning Availability of 2-5 umbilical cord blood units that are at least a 4/6 HLA match No active CNS disease No primary or grade 3 or 4 myelofibrosis PATIENT CHARACTERISTICS: Age Any age Performance status Karnofsky 70-100% (for patients 16 years of age and older) Lansky 50-100% (for patients under 16 years of age) Life expectancy At least 3 months Hematopoietic See Disease Characteristics Hepatic ALT/AST less than 4 times normal Bilirubin less than 2.0 mg/dL (unless due to hepatic infiltration by primary malignancy) Renal Creatinine clearance greater than 40 mL/min Cardiovascular Shortening fraction or ejection fraction greater than 40% of normal value for age by echocardiogram or radionuclide scan Pulmonary FVC and FEV_1 greater than 60% of predicted DLCO greater than 60% of predicted (adult patients) Clearance by pulmonologist required if patient cannot perform pulmonary function tests Other Not pregnant or nursing No uncontrolled active infection (viral, bacterial, or fungal) HIV negative PRIOR CONCURRENT THERAPY: Biologic therapy See Disease Characteristics More than 3 months since prior autologous stem cell transplantation Chemotherapy See Disease Characteristics At least 4 weeks since prior chemotherapy Endocrine therapy Not specified Radiotherapy Not specified Surgery Not specified Other Recovered from prior therapy No other concurrent investigational agents that would preclude study participation or increase risk to patient Investigational diagnostic procedures allowed