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 (PI) prior to enrollment and at the Patient Care Conference or equivalent group such as the pediatric leukemia board as an alternative * Patients in whom adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the PI prior to enrollment; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with PI approval * Very high risk pediatric/young adult patients with acute myeloid leukemia (AML): Patients =\< 25 years, however, are eligible with (M2 marrow) with =\< 25% blasts in marrow after having failed one or more cycles of chemotherapy; this group of patients will be analyzed separately * Acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: * All patients must be in CR as defined by \< 5% blasts by morphology; flow cytometry in a representative bone marrow sample with cellularity \>= 15% for age; patients who do not have high-risk disease (high risk CR1, greater than one cycle to obtain CR or \>= CR2) must be discussed with the PI prior to enrollment and at the Patient Care Conference or equivalent group such as the pediatric leukemia board as an alternative * Patients in whom adequate marrow/biopsy specimens cannot be obtained to determine remission status by morphologic assessment, but have fulfilled criteria of remission by flow cytometry, recovery of peripheral blood counts with no circulating blasts, and/or normal cytogenetics (if applicable) may still be eligible; reasonable attempts must be made to obtain an adequate specimen for morphologic assessment, including possible repeat procedures; these patients must be discussed with the principal investigator Ann Dahlberg prior to enrollment; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with PI approval * Chronic myelogenous leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to imatinib mesylate * Advanced myelofibrosis * Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], RAEB in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% by a representative bone marrow aspirate morphology * Lymphoblastic lymphoma, Burkitt's lymphoma, and other high-grade non-Hodgkin lymphoma (NHL) after initial therapy if stage III/IV in first partial response (PR1) or after progression if stage I/II \< 1 year; stage III/IV patients are eligible after progression in CR/PR * Chronic lymphocytic leukemia /small lymphocytic lymphoma (CLL/SLL), marginal zone B-cell lymphoma, lymphoplasmacytic lymphoma or follicular lymphoma that have progressed after at least two different prior therapies; patients with bulky disease (nodal mass greater than 5 cm) should be considered for debulking chemotherapy before transplant; these patients must be presented at primary care center (PCC) prior to enrollment, given potential competing eligibility on autotransplant protocols * Mantle-cell lymphoma, prolymphocytic leukemia: Eligible after initial therapy in \>= CR1 or \>= PR1 * Large cell NHL \> CR2/\> second partial response (PR2): * Patients in CR2/PR2 with initial short remission (\< 6 months) are eligible * These patients must be presented at PCC prior to enrollment, given potential competing eligibility on autotransplant protocols * Multiple myeloma beyond PR2: Patients with chromosome 13 abnormalities, first response lasting less than 6 months, or beta-2 microglobulin \> 3 mg/L, may be considered for this protocol after initial therapy * Performance status score: Karnofsky (for adults) \>= 70% or Eastern Cooperative Oncology Group (ECOG) 0-1 or Lansky (for children) \>= 50% * Creatinine \< 2.0 mg/dL (for adults) or creatinine clearance \> 60 ml/min (for children) * Patients with clinical or laboratory evidence of liver disease will be evaluated for the cause of liver disease, its clinical severity in terms of liver function, histology, and the degree of portal hypertension; patients with fulminant liver failure, cirrhosis with evidence of portal hypertension or bridging fibrosis, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, or correctable hepatic synthetic dysfunction 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, and symptomatic biliary disease will be excluded * Diffusion capacity for carbon monoxide corrected (DLCOcorr) \> 50% normal or a pediatric patient who is unable to perform pulmonary function tests (PFTs) but has adequate pulmonary function * Left ventricular ejection fraction \> 45% or shortening fraction \> 26% Exclusion Criteria: * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 month) invasive fungal infection without interdisciplinary (ID) consult and approval * History of human immunodeficiency virus (HIV) infection * Pregnant or breastfeeding * Chemotherapy refractory large cell and high grade NHL (i.e., progressive disease after \> 2 salvage regimens) * Patients with history of prior myeloablative transplant containing full dose TBI (greater than 8 gray \[Gy\]) will not be eligible for Regimen A; however, they may still enroll on Regimen B if they otherwise meet inclusion and exclusion criteria * Any prior myeloablative transplant within the last 6 months * Patients \>= 45 years: comorbidity score of 5 or higher * Patients who have received Y-90 ibritumomab (Zevalin) or I-131 tostumomab (Bexxar), as part of their salvage therapy are not eligible for Regimen A