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