Umbilical Cord Blood Transplant, Cyclophosphamide, Fludarabine, and Total-Body Irradiation in Treating Patients With Hematologic Disease

Inclusion Criteria: * GRAFT CRITERIA: * UCB units will be selected according to current umbilical cord blood graft selection algorithm; one or 2 UCB units may be used to achieve the required cell dose * The UCB graft is matched at 4-6 human leukocyte antigen (HLA)-A, B, DRB1 antigens with the recipient; this may include 0-2 antigen mismatches at the A or B or DRB1 loci; unit selection based on cryopreserved nucleated cell dose and HLA-A,B, 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 4-6 antigen match to the recipient * Age and Disease Criteria: * High-dose TBI regimen: 6 months to =\< 45 years * Middle-intensity TBI regimen: 6 months to =\< 65 years * Conditioning regimen selection should be based on the underlying disease, presence of minimum residual disease (MRD), age, co-morbidities, and attending physician * Acute myeloid leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: * All patients must be in complete remission (CR) as defined by hematologic recovery and \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age; patients who do not have high-risk features (for example preceding myelodysplastic syndrome \[MDS\], high-risk cytogenetics, \>= 2 cycles to obtain CR, erythroblastic or megakaryocytic leukemia or \>= CR2) must be discussed with the principal investigator …