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 mesylate, dasatinib, nilotinib) or predicted to be so based upon clinical course or abl kinase domain mutation analysis; or in accelerated phase or blast crisis; * High intermediate to high international prognostic score myelodysplasia; * Non-Hodgkin lymphoma (NHL)/Hodgkin lymphoma (HL)/other lymphoproliferative diseases resistant/refractory to standard therapies and for whom an autologous transplant is considered to be inappropriate (e.g. bone marrow involvement, chemotherapy refractory disease, prior transplant); * Chronic lymphocytic leukemia (CLL) resistant/refractory to standard therapies (e.g. fludarabine) or high risk cytogenetics/fluorescence in situ hybridization (FISH) (e.g. 17p-); * Myeloproliferative disorders with progressive disease or cytopenias or clinical symptoms refractory to standard therapy (e.g. hypomethylating agents) * Relapsed or refractory multiple myeloma after (or not eligible for) high dose chemotherapy/autologous hematopoietic stem cell rescue and following salvage therapy with thalidomide, lenalidomide or bortezomib/other Food and Drug Administration (FDA)-approved multiple myeloma salvage therapies; * Other hematologic malignancies/disorders with anticipated 2 year survival \< 20%, as established by available data bases, medical literature and the documented consensus of the Hematologic Malignancies Tumor Study Group * Patients must be an allogeneic HSCT candidate but have no standard donor (matched related donor \[MRD\], human leukocyte antigen \[HLA\]-matched unrelated donor \[MUD\] or single UCB unit of appropriate size and HLA type) available * Patients must have available UCB unit(s) * Patients considered for Arm 2 must not be eligible for Arm 1 and must have an HLA-haploidentical sibling, parent, child, or other relative (uncle, aunt, first cousin, niece or nephew) who meets donor requirements as outlined in Donor Eligibility criteria * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 * Left ventricular (LV) ejection fraction \>= 50% * Diffusion capacity of carbon monoxide (DLCO) corrected for hemoglobin \> 60% * Total bilirubin within normal institutional limits unless the patient has Gilbert's disease * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal (ULN) * Measured or estimated creatinine clearance \> 50 ml/min * Hematopoietic stem cell co-morbidity index =\< 2 * There must be a negative pregnancy test for women of childbearing potential within 1 week of therapy; there must be willingness to avoid pregnancy and undergo counseling about contraceptive techniques throughout the course of treatment * There must be no uncontrolled infections or active acute or chronic illnesses such as diabetes, angina/myocardial ischemia, cardiac arrhythmia, venous thrombosis/embolism, cerebrovascular disease, seizure disorder, psychiatric illness or other intercurrent illness that is not well controlled or is anticipated to be difficult to control during the proposed therapy * The patient must be aware of the high risk and experimental nature of the treatment and provide informed consent * The patient must have clearance for HSCT after psychosocial evaluation * The patient must have adequate insurance or other support to meet the anticipated financial burden imposed by the costs of therapy * DONOR (for allogeneic lymphocytes, Arm 2 only): Relative (parent, child, sibling, first cousin, uncle aunt, nephew, niece) with appropriate HLA match (\>= 3/6 HLA A, B, DR match) * DONOR (for allogeneic lymphocytes, Arm 2 only): Age \>= 18 years old * DONOR (for allogeneic lymphocytes, Arm 2 only): Normal hemogram; potential donors not having a normal hemogram may be utilized at the discretion of the Principal Investigator * DONOR (for allogeneic lymphocytes, Arm 2 only): Not pregnant or lactating * DONOR (for allogeneic lymphocytes, Arm 2 only): Not human immunodeficiency virus (HIV)-1, HIV-2, hepatitis C (HCV), Hepatitis B core or human T-lymphotropic virus (HTLV)-I/II seropositive; hepatitis B surface antigen (HB Sag)(-); must meet other infectious disease screening criteria utilized by New Brunswick Affiliated Hospital (NBAH) Blood Center * DONOR (for allogeneic lymphocytes, Arm 2 only): No uncontrolled infections, other medical or psychological/social conditions, or required medications that might increase the likelihood of patient or donor adverse effects or poor outcomes * DONOR (for allogeneic lymphocytes, Arm 2 only): Meet other blood bank criteria for blood product donation (as determined by NBAH Blood Center screening history) * DONOR (for allogeneic lymphocytes, Arm 2 only): Donors must be informed of the investigational nature of this study, understand the requirements, potential benefits and potential risks of the experimental treatment, and give written informed consent in accordance with institutional and federal guidelines Exclusion Criteria: * Prior extensive radiation therapy that the radiation oncologist feels precludes additional TBI * Patients with known human immunodeficiency virus (HIV) are excluded due to side effects of the therapy on the immune system * Patients with known active central nervous system (CNS) disease will be excluded from this clinical trial because they often develop progressive neurologic dysfunction unresponsive to HSCT therapy