Infusion of Off-the-Shelf Expanded Cord Blood Cells to Augment Cord Blood Transplant in Patients With Hematologic Malignancies

Inclusion Criteria:

* Acute myeloid leukemia:

  * High risk complete response (CR)1 as evidenced by preceding myelodysplastic syndromes (MDS), high risk cytogenetics (for example, monosomy 5 or 7, or as defined by referring institution treatment protocol), \>= 2 cycles to obtain CR, erythroblastic or megakaryocytic leukemia; \>= CR2
  * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15%
  * 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, Colleen Delaney prior to enrollment
* Acute lymphoblastic leukemia:

  * High risk CR1 \[for example, but not limited to: t(9;22), t(1;19), t(4;11) or other mixed lineage leukemia (MLL) rearrangements, hypodiploid\]
  * Greater than 1 cycle to obtain CR
  * \>= CR2
  * All patients must be in CR as defined by hematologic recovery and \< 5% blasts by morphology within the bone marrow and a cellularity of \>= 15%
  * 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, Colleen Delaney prior to enrollment
* 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
* Myelodysplasia (MDS) International Prognostic Scoring System (IPSS) intermediate 2 (Int-2) or high risk (i.e., refractory anemia with excess myeloblasts \[RAEB\], refractory anemia with excess blasts in transformation \[RAEB-T\]) or refractory anemia with severe pancytopenia or high risk cytogenetics; blasts must be \< 10% by a representative bone marrow aspirate morphology
* Karnofsky (\>= 16 years old) \>= 70%
* Lansky (\< 16 years old) \>= 50%
* Calculated creatinine clearance must be \> 60 mL and serum creatinine =\< 2 mg/dL (adults)
* Calculated creatinine clearance must be \> 60 mL/min (children \< 18 years old)
* Total serum bilirubin must be \< 3 mg/dl
* Transaminases must be \< 3 x the upper limit of normal
* Diffusion capacity of the lung for carbon monoxide (DLCO) corrected \> 50% normal
* For pediatric patients unable to perform pulmonary function tests, oxygen (O2) saturation \> 92% on room air
* Left ventricular ejection fraction \> 45% OR shortening fraction \> 26%
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Uncontrolled viral or bacterial infection at the time of study enrollment
* Active or recent (prior 6 month) invasive fungal infection without infectious disease (ID) consult and approval
* History of human immunodeficiency virus (HIV) infection
* Pregnant or breastfeeding
* If =\< 18 years old, prior myeloablative transplant within the last 6 months
* If \> 18 years old prior myeloablative allotransplant or autologous transplant
* Extensive prior therapy including \> 12 months alkylator therapy or \> 6 months alkylator therapy with extensive radiation