Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematological Cancer or Other Disease

DISEASE CHARACTERISTICS: * Diagnosis of advanced hematologic malignancy or other disease not curable by conventional chemotherapy, including any of the following: * Acute myeloid leukemia in complete remission (CR)\* (as defined by hematologic recovery, \< 5% blasts in the bone marrow by morphology, and a cellularity of \> 15%), meeting one of the following criteria: * In first complete remission (CR1) AND has high-risk disease as evidenced by any of the following: * Preceding myelodysplastic syndromes (MDS) * High-risk cytogenetics (e.g., monosomy 5 or 7, or as defined by referring institution treatment protocol) * Required \> 2 courses of therapy to obtain CR * Erythroblastic or megakaryocytic leukemia * In second CR (CR2) or beyond * Acute lymphoblastic leukemia in CR\* (as defined by hematologic recovery, \< 5% blasts in the bone marrow by morphology, and a cellularity of \> 15%), meeting one of the following criteria: * In CR1 AND has high-risk disease as evidenced by any of the following: * t(9;22), t(1;19), t(4;11), or other MLL rearrangements * Hyplodiploid * Required \> 1 course of therapy to obtain CR * Beyond CR2 * Chronic myelogenous leukemia (CML) * All types are allowed (except refractory blast crisis CML) * Patients in chronic phase CML must have failed or been intolerant to prior imatinib mesylate (Gleevec) or other tyrosine kinase inhibitors * MDS * Any subtype allowed (including refractory anemia \[RA\]) * Severe pancytopenia or complex cytogenetics * Blasts must be \< 5% (if blasts are ≥ 5%, pre-transplant induction therapy is required to reduce blast count to \< 5%) * Large cell lymphoma, Hodgkin lymphoma, or multiple myeloma, meeting one of the following criteria: * Chemotherapy-sensitive disease that has failed prior therapy * Patients with large cell lymphoma or Hodgkin lymphoma must not have progressive disease during salvage therapy (stable disease allowed provided it is non-bulky) * Ineligible for an autologous stem cell transplant * Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, or follicular lymphoma that has progressed after ≥ 2 prior therapies * Patients with bulky disease should be considered for debulking chemotherapy prior to transplant * Patients with refractory disease are eligible provided disease is non-bulky AND an estimated tumor doubling time is ≥ 1 month * Lymphoplasmacytic lymphoma, mantle cell lymphoma, or prolymphocytic leukemia * Chemotherapy-sensitive disease that was previously treated with initial therapy * Patients with mantle cell lymphoma must not have progressive disease during salvage therapy (stable disease allowed provided it is non-bulky) * Mycosis fungoides and Sezary syndrome * Bone marrow failure syndromes, except for Fanconi anemia * Myeloproliferative syndromes NOTE: \*Patients for whom adequate marrow/biopsy specimens can not be obtained to determine remission status by morphologic assessment must have fulfilled criteria of remission (\< 5% blasts by flow cytometry and recovery of peripheral blood counts with no circulating blasts) * Ineligible for autologous stem cell transplant due to any of the following: * Prior autologous stem cell transplant * Inadequate autologous stem cell harvest * Inability to withstand a myeloablative preparative regimen * Clinically aggressive/high-risk disease * No evidence of progressive disease by imaging modalities or biopsy (persistent PET scan activity allowed provided there are no CT scan changes indicating progression) * Acute leukemia that is refractory, persistent, or relapsed (defined as \> 5% blasts in normocellular bone marrow) allowed provided patient was rendered aplastic either by induction chemotherapy or radioimmunoconjugated monoclonal antibody therapy * Patients with stable disease are eligible provided the largest residual nodal mass is approximately \< 5 cm (largest residual mass must represent a 50% reduction and be approximately \< 7.5 cm for patients who have responded to prior therapy) * No active CNS malignancy * Umbilical cord blood (UCB) donor available * UCB graft matched at 4/6 HLA-A, -B, and -DRB1 antigens with the recipient * May include 0-2 antigen mismatches at the A, B, or DRB1 loci * Unit selection based on cryopreserved nucleated cell dose and HLA-A, -B, and -DRB1 using intermediate resolution A, B antigen and DRB1 allele typing * If 2 UCB units are required to reach the target cell dose, each unit must be a 3/6 HLA-A, -B, and -DRB1 antigen match to each other, as well as a 4/6 antigen match to the recipient * No 5-6/6 HLA-A, -B, and -DRB1 matched sibling donor available PATIENT CHARACTERISTICS: * Karnofsky performance status (PS) 60-100% OR Lansky PS 50-100% * Creatinine ≤ 2.0 mg/dL (adults) OR creatinine clearance \> 40 mL/min (pediatrics) * Adult patients with a creatinine \> 1.2 mg/dL or a history of renal dysfunction must have an estimated creatinine clearance of \> 40 mL/min * Not pregnant or nursing * Negative pregnancy test * LVEF ≥ 35% * DLCO \> 30% predicted * No requirement for O\_2 * No decompensated congestive heart failure * No uncontrolled arrhythmia * None of the following liver diseases or conditions: * Fulminant liver failure * Cirrhosis with evidence of portal hypertension or bridging fibrosis * Alcoholic hepatitis * Esophageal varices * History of bleeding esophageal varices, hepatic encephalopathy, or correctable hepatic synthetic dysfunction as evidenced by prolongation of the prothrombin time * Ascites related to portal hypertension * Bacterial or fungal abscess * Biliary obstruction * Chronic viral hepatitis with total serum bilirubin \> 3 mg/dL * Symptomatic biliary disease * Recent mold infection (e.g., Aspergillus) allowed provided patient received ≥ 30 days of appropriate treatment AND infection is controlled and cleared by Infectious Disease * No evidence of HIV infection or known HIV-positive serology * No uncontrolled viral or bacterial infection PRIOR CONCURRENT THERAPY: * See Disease Characteristics * At least 3 months since prior myeloablative stem cell transplantation