Treosulfan, Fludarabine Phosphate, and Total-Body Irradiation in Treating Patients With Hematological Cancer Who Are Undergoing Umbilical Cord Blood Transplant

Inclusion Criteria: * Acute myeloid leukemia/acute lymphoblastic leukemia, including biphenotypic acute leukemia or mixed-lineage leukemia: Must have \< 5% morphologic marrow blasts in an evaluable marrow sample (\> 25% of normal cellularity for age collected less than one month prior to start of conditioning; patients in which 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 prior to enrollment; patients persistently aplastic for greater than one month since completing last chemotherapy are also eligible with principal investigator (PI) approval * Myelodysplastic syndrome (MDS): Any 2001 World Health Organization (WHO) classification subtype; refractory anemia with excess blasts (RAEB)-2 patients may proceed directly to transplant, but may also be considered for induction chemotherapy before transplant; patients with \>= 20% morphologic marrow blasts require induction therapy to reduce morphologic marrow blasts below 5% before transplant * Chronic myelogenous leukemia: All types, except refractory blast crisis; chronic phase patients must have failed or been intolerant to Gleevec or other tyrosine kinase inhibitors * Patients =\< 50 must have performance status score: Karnofsky (for adults) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1; Lansky (for children) score \>= 50 * Patients \> 50 must have Karnofsky performance score \>= 70 or ECOG 0-1 and comorbidity index \< 5 * Adequate cardiac function defined as absence of decompensated congestive heart failure or uncontrolled arrhythmia AND left ventricular ejection fraction \>= 35% OR fractional shortening \> 22% * Adequate pulmonary function defined as absence of oxygen (O2) requirements and one of the following: * Diffusion lung capacity for carbon monoxide (DLCO) corrected \>= 70% mm Hg * DLCO corrected between 60% - 69% mm Hg and partial pressure of oxygen (pO2) \>= 70 mm Hg * DLCO corrected between 50% - 59% mm Hg and pO2 \>= 80 mm Hg * Pediatric patients unable to perform pulmonary function tests must have O2 saturation \> 92% on room air; may not be on supplemental oxygen * Adequate hepatic function; 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 * Adequate renal function defined as creatinine =\< 2.0 mg/dl (adults) or estimated creatinine clearance \> 40 ml/min (pediatrics); all adults with a creatinine \> 1.2 or a history of renal dysfunction must have estimated creatinine clearance \> 40 ml/min * If recent mold infection, e.g., Aspergillus, must be cleared by infectious disease to proceed * Prior hematopoietic cell transplant: must be \>= 3 months after previous transplant * DONOR: Human leukocyte antigen (HLA) matching: * Minimum requirement: The cord blood (CB) graft(s) must be matched at a minimum at 4/6 HLA-A, B, DRB1 loci with the recipient; therefore 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution A, B antigen and DRB1 allele typing for determination of HLA-match is allowed * HLA-matching determined by high resolution typing is allowed per institutional guidelines as long as the minimum criteria (above) are met * DONOR: Selection of two CB units is mandatory when a single cord blood unit does not meet the following criteria in the table below * Match grade * 6/6 * Single unit allowed for total nucleated cell (TNC) dose \>= 2.5 x 10\^7/kg * 5/6, 4/6 * Single unit allowed for TNC dose \>= 4.0 (+/- 0.5) x 10\^7/kg * If two CB units are used, the total cell dose of the combined units must be at least 3.0 x 10\^7 TNC per kilogram recipient weight based on pre-cryopreservation numbers, with each CB unit containing a MINIMUM of 1.5 x 10\^7 TNC/kg * DONOR: The minimum recommended CD34/kg cell dose should be 2 x 10\^5 CD34/kg, total dose from a single or combined double * DONOR: The unmanipulated CB unit(s) will be Food and Drug Administration (FDA) licensed or will be obtained under a separate investigational new drug (IND), such as the National Marrow Donor Program (NMDP) Protocol 10-CBA conducted under BB IND-7555 or another IND sponsored by (1) a participating institution or (2) an investigator at FHCRC or one of the participating institutions * DONOR (FHCRC only): Up to 5% of the unmanipulated cord blood product (s), when ready for infusion, may be withheld for research purposes as long as thresholds for infused TNC dose are met; these products will be used to conduct studies involving the kinetics of engraftment and immunobiology of double cord transplantation Exclusion Criteria: * Pregnancy or breastfeeding * Evidence of human immunodeficiency virus (HIV) infection or known HIV positive serology * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 month) invasive fungal infection without infectious diseases (ID) consult and approval * Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy and/or cranial radiation prior to initiation of conditioning (day -6) * DONOR: Any cord blood units with \< 1.5 x 10\^7 total nucleated cells per kilogram recipient weight * DONOR: Any cord blood units that have not passed donor screening for infectious disease markers as recommended by NMDP will not be used unless a waiver is signed by the clinical attending allowing use of CB unit. Cord blood units are presumed to be cytomegalovirus (CMV) negative regardless of serologic testing due to passive transmission of maternal CMV antibodies