Single or Double Donor Umbilical Cord Blood Transplant in Treating Patients With High-Risk Hematologic Malignancies

Inclusion Criteria: * Patients with histologically proven hematologic malignancy with anticipated 2 year survival \< 20% with standard therapy; patients age \<18 are excluded by virtue of the policies and procedures of the allogeneic hematopoietic stem cell transplant (HSCT) program (Cancer Institute of New Jersey \[CINJ\]/Robert Wood Johnson University Hospital \[RWJUH\] is not an approved Pediatric Transplant Center); patients \> age 65 are generally not considered candidates for experimental unrelated allogeneic HSCT, as utilized in this study by virtue of the anticipated delayed immune reconstitution, high risk of GVHD, and known negative impact of age on outcomes * Patients eligible for this trial will have high risk diseases that include, but are not limited to: * Acute myeloid leukemia (AML) in second complete remission (CR2) or greater or early relapse with \< 5% marrow blasts and no circulating blasts * AML in first complete remission (CR1) with high risk cytogenetics (complex, monosomy 5, monosomy 7, 11q23 (not t(9;11)), t(6;9), chromosome 3, monosomy phenotype and other karyotypes estimated to have =\< 20% disease free survival at 3 years) or secondary/transformed AML without favorable cytogenetics; * Acute lymphoblastic leukemia (ALL) with t(9;22), 11q23 abnormality or early relapse (\< 5% marrow blasts) or CR2 or greater; * Chronic myeloid leukemia (CML) resistant/refractory to all commercially available Abelson (abl) kinase inhibitors (e.g. imatinib me…