Infusion of Expanded Cord Blood Cells in Addition to Single Cord Blood Transplant in Treating Patients With Acute Leukemia, Chronic Myeloid Leukemia, or Myelodysplastic Syndromes

Inclusion Criteria: * Patients 10 to 65 years old with a hematologic malignancy in need of hematopoietic cell transplant who are \> 30 kg and without a suitable related donor * Patient must have hematologic malignancy that meets institutional eligibility requirements for cord blood transplant * Malignancies included are: * Acute leukemia, including acute myeloid leukemia (AML), biphenotypic acute leukemia or mixed-lineage leukemia, acute lymphoblastic leukemia (ALL); all patients must be in complete response (CR) as defined by \< 5% blasts by morphology/flow cytometry in a representative bone marrow sample with adequate cellularity to assess remission status * Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate (Int)-2 or high risk (i.e., refractory anemia with excess blasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEBt\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% in a representative bone marrow aspirate * Chronic myeloid leukemia excluding refractory blast crisis; to be eligible in first chronic phase (CP1) patient must have failed or be intolerant to tyrosine kinase inhibitor therapy * High dose TBI regimen: 10 to =\< 45 years * Intermediate intensity regimen: 10 to =\< 65 years * Patients 10 to =\< 45 years: Lansky (\< 16 years old) or Karnofsky (\>= 16 years old) \>= 70 or Eastern Cooperative Oncology Group (ECOG) 0-1 * Patients \> 45 to =\< 65 years: Karnofsky \>= 70 or ECOG 0-1 and non-age adjusted comorbidity index =\< 5 * Adults: Calculated creatinine clearance must be \> 60 mL and serum creatinine =\< 2 mg/dL * Children (\< 18 years old): Calculated creatinine clearance must be \> 60 mL/min * Total serum bilirubin must be \< 3 mg/dL unless the elevation is thought to be due to Gilbert's disease or hemolysis * Transaminases must be \< 3 x the upper limit of normal per reference values of treating institution * Carbon monoxide diffusing capability (DLCO) corrected \>= 60% normal (may not be on supplemental oxygen) * For pediatric patients unable to perform pulmonary function tests, O2 saturation \> 92% on room air * Left ventricular ejection fraction \>= 50% OR * Shortening fraction \> 26% * Ability of participant or legally authorized representative to understand and the willingness to sign a written informed consent form * DONOR: Minimum requirement: The cord blood (CB) unit must be matched at a minimum at 4/6 HLA-A, B antigens and DRB1 allele with the recipient; therefore, 0-2 mismatches at the A or B or DRB1 loci based on intermediate resolution at HLA-A, B and high resolution allele level typing at HLA- DRB1 are allowed * DONOR: Institutional guidelines for HLA-match may be followed as long as the minimum criteria for HLA-matching as above are met * DONOR: The CB unit selected for transplant must have a MINIMUM of 2.5 x 10\^7 TNC/kg * DONOR: The minimum recommended CD34/kg cell dose is 1.7 x 10\^5 CD34/kg * DONOR: A backup unit must be identified and reserved prior to the start of the treatment plan for possible infusion in the unlikely event of poor post-thaw viability of the primary CB unit. A suitable back up unit will be considered, as follows: * Must be matched at a minimum at 4/6 HLA-A, B, DRBl loci with the recipient. Therefore 0-2 mismatches at the A or B or DRBl loci based on intermediate resolution A, B antigen and DRBl allele typing for determination of HLA-match is allowed (Fred Hutch Protocol 2010). * Must contain a MINIMUM of 1.5 x 10\^7 TNC/kg to ensure the same requirement we use for a standard double CBT per CB selection guideline (Fred Hutch Protocol 2010). Exclusion Criteria: * Uncontrolled viral or bacterial infection at the time of study enrollment * Active or recent (prior 6 month) invasive fungal infection unless cleared by infectious disease (ID) consult * History of human immunodeficiency virus (HIV) infection * Pregnant or breastfeeding * Prior allogeneic transplant * Central nervous system (CNS) leukemic involvement not clearing with intrathecal chemotherapy; diagnostic lumbar puncture is to be performed * \< 30 kg