Donor Umbilical Cord Blood Transplant After Fludarabine Phosphate, Cyclophosphamide, and Total-Body Irradiation in Treating Patients With High-Risk Hematologic Cancer

Inclusion Criteria: * Disease Criteria * Acute Leukemias: Must be in remission by morphology (\<5% blasts). Note cytogenetic relapse or persistent disease without morphologic relapse is acceptable. Also a small percentage of blasts that is equivocal between marrow regeneration vs. early relapse are acceptable provided there are no associated cytogenetic markers consistent with relapse. (See exclusion criteria for more detailed definition) * Acute myeloid leukemia: high risk CR1 (as evidenced by preceding myelodysplastic syndrome (MDS), high risk cytogenetics such as those associated with MDS or complex karyotype, \> 2 cycles to obtain complete remission (CR) or erythroblastic and megakaryocytic); second or greater CR. * Acute lymphoblastic leukemia/lymphoma: high risk CR1 as evidenced by high risk cytogenetics (e.g. t(9;22, t(1;19), t(4;11), other MLL rearrangements) or \> 1 cycle to obtain CR; second or greater CR. * Burkitt's lymphoma in CR2 or subsequent CR * Natural Killer cell malignancies * Myeloproliferative syndromes/diseases: Chronic myelogenous leukemia (CML) in chronic or accelerated phase but patients must have failed or been intolerant to Imatinib mesylate. EXCLUDED: CML in refractory blast crisis, myelofibrosis, polycythemia vera, and essential thrombocytosis. * Myelodysplastic Syndrome: any subtype including refractory anemia (RA) if severe pancytopenia, high risk complex cytogenetics or International Prognostic Scoring System (IPSS) ≥ intermediate-2 (Int-2). Blasts must be less than 5%. If 5% or more requires therapy pre-transplant to reduce blast count to ≤5%. Patients who receive single agent 5-azacytidine, decitabine or immunomodulating drugs are eligible. * Large-cell lymphoma, Hodgkin's lymphoma and multiple myeloma: patients with chemotherapy sensitive disease that has failed or who are ineligible for an autologous transplant. Patients are eligible for umbilical cord blood (UCB) transplantation if there is no evidence of progressive disease by imaging modalities and/or biopsy. Persistent PET activity, though possibly related to lymphoma, IS NOT an exclusion criterion in the absence of computated tomography (CT) changes in size indicating progression. Large-cell and Hodgkin's lymphoma that is progressive on salvage therapy is NOT eligible. Patients with stable disease are eligible for transplantation if the largest residual nodal mass is \< 5 cm (approximately). For patients who have responded to preceding therapy, the largest residual mass must represent a 50% reduction and be \< 7.5 cm (approximately). * Chronic lymphocytic leukemia/small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, follicular lymphoma which have progressed after at least two prior therapies. Patients with bulky disease should be considered for debulking chemotherapy before transplant. Patients with refractory disease are eligible, unless has bulky disease and an estimated tumor doubling time of less than one month. Patients with stable disease are eligible for transplantation if the largest residual nodal mass is \< 5 cm (approximately). For patients who have responded to preceding therapy, the largest residual mass must represent a 50% reduction and be \< 7.5 cm (approximately). * Lymphoplasmacytic lymphoma, mantle-cell lymphoma, prolymphocytic leukemia are eligible after initial therapy if chemotherapy sensitive. Mantle-cell lymphoma that is progressive on salvage therapy is NOT eligible. Patients with stable disease are eligible for transplantation if the largest residual nodal mass is \< 5 cm (approximately). For patients who have responded to preceding therapy, the largest residual mass must represent a 50% reduction and be \< 7.5 cm (approximately). * Umbilical Cord Blood Graft Selection - Two UCB units will be compose the graft, and each unit must be a 4-6 HLA-A, B, DRB1 antigen match to each other, as well as a 4-6 antigen match to the recipient. The combined cryopreserved nucleated cell dose of the 2 units must be ≥ 3 X 10\^7/kg with each unit having a minimum cell dose of 1.5 X 10\^7/kg. UCB units will be selected according to a common umbilical cord blood graft selection algorithm * Performance Status - adequate performance status defined as Karnofsky score ≥ 60 * Age 18 to 70 years of age; patients ≥ 70 but ≤ 75 years are eligible if the co-morbidity score is ≤ 2 * Organ Function * Cardiac: Left ventricular ejection fraction \> 35%; absence of decompensated congestive heart failure; absence of uncontrolled arrhythmia * Pulmonary: DLCO \> 30% of predicted; absence of O2 requirements * Hepatic: ALT, AST, alkaline phosphatase and bilirubin \< 5 x upper limit of normal * Renal: Creatinine ≤ 2 mg/dl (patients with a creatinine \> 1.2 or history of renal dysfunction must have calculated glomerular filtration rate (GFR) \> 40 mL/min/1.73m2) * If recent mold infection e.g. Aspergillus - must have minimum of 30 days of appropriate treatment before transplant and infection controlled and be cleared by Infectious Disease. * The following conditions must be met: * If prior myeloablative autologous transplant, must be \> 3 months but ≤ 12 months from transplant OR have received at least 2 cycles of multi-agent or highly immunosuppressive chemotherapy (i.e. induction for acute leukemia) within the 3 months preceding this study. OR * If neither prior myeloablative autologous transplant ≤ 12 months from transplant nor have received at least 2 cycles of multi-agent or highly immunosuppressive chemotherapy (i.e. induction for acute leukemia) within the 3 months preceding this study, patients are eligible as long as they receive equine anti-thymocyte globulin as part of the conditioning regimen. Exclusion Criteria: * Patients who have an available, medically suitable, 5-6/6 HLA-A, B, DRB1 matched sibling donor * Patients who are eligible for autologous transplantation * Prior allogeneic transplant * Acquired or inherited bone marrow failure syndromes such as aplastic anemia and Fanconi anemia * Pregnant or breast feeding * Evidence of HIV infection or known HIV positive serology * Current uncontrolled serious infection * Active central nervous system malignancy