Fludarabine Phosphate, Melphalan, Total-Body Irradiation, Donor Stem Cell Transplant in Treating Patients With Hematologic Cancer or Bone Marrow Failure Disorders

Inclusion Criteria: * Diagnosis of a histology documented hematologic malignancy or marrow disorder * Bone marrow failure disorders and other non-malignant hematologic or immunologic disorders: * Acquired bone marrow failure disorders include aplastic anemia, paroxysmal nocturnal hemoglobinuria (PNH): * Primary allogeneic hematopoietic stem cell transplantation (HSCT) is appropriate for selected patients with severe aplastic anemia; however, patients with aplastic anemia must have failed at least one cycle of standard immunosuppressive therapy with calcineurin inhibitor plus anti-thymocyte globulin (ATG) if a fully-matched donor is not available * Patients with PNH must have a history of thrombosis related to PNH * Hereditary bone marrow failure disorders include Fanconi anemia or related chromosomal breakage syndrome dyskeratosis congenita, Diamond-Blackfan anemia, Shwachman-Diamond syndrome, Kostmann syndrome, congenital amegakaryocytic thrombocytopenia: * Fanconi anemia or related chromosomal breakage syndrome: positive chromosome breakage analysis using diepoxybutane (DEB) or mitomycin C if applicable * Dyskeratosis: diagnosis is supported by using either telomerase reverse transcriptase (TERC) gene mutation in autosomal dominant Dyskeratosis Congenita or Xlinked DKC1 gene mutation * Other non-malignant hematologic or immunologic disorders that require transplantation * Quantitative or qualitative congenital platelet disorders (including but not limited to congenital amegakaryocytopenia, absent-radii syndrome, Glanzmann's thrombasthenia) * Quantitative or qualitative congenital neutrophil disorders (including but not limited to chronic granulomatous disease, congenital neutropenia) * Congenital primary immunodeficiencies (including but not limited to Severe Combined Immunodeficiency Syndrome, Wiskott-Aldrick syndrome, CD40 ligand deficiency, T-cell deficiencies) * Acute leukemias: * Subjects must be ineligible for conventional myeloablative transplantation; * Resistant or recurrent disease after at least one standard combination chemotherapy regime or first remission patients at high risk of relapse OR First remission patients at high risk of relapse: * Acute myeloid leukemia (AML)- antecedent myelodysplastic syndrome, secondary AML, high risk cytogenetic abnormalities or normal cytogenetics with high-risk molecular features (e.g. Flt3-ITD mutation, mixed-lineage leukemia \[MLL\], wildtype NPM1); * Acute lymphocytic leukemia (ALL)- high or standard risk ALL * Chronic Myeloid Leukemia (CML): * Chronic phase (intolerant or unresponsive to imatinib and/or other tyrosine kinase inhibitors), second chronic phase or accelerated phase who are ineligible for conventional myeloablative transplantation * Myeloproliferative and myelodysplastic syndromes (MDS): * Myelofibrosis (with/without splenectomy) with intermediate to high risk features * Advanced polycythemia vera not responding to standard therapy * MDS with an international prostate symptom score (IPSS) score of Int-2 or higher * MDS with lower IPSS scores Int-1 or less with severe clinical features such as severe neutropenia or thrombocytopenia or high risk chromosome abnormalities such as monosomy 7 * Secondary massively parallel signature sequencing (MPSS) with any IPSS scores * Chronic myelomoncytic leukemia * Lymphoproliferative disease: * Chronic lymphocytic leukemia (CLL), low-grade non-Hodgkin lymphoma (NHL) (recurrent or persistent) fludarabine refractory or with less than 6 months duration of complete response (CR) between courses of conventional therapy * Multiple myeloma, progressive disease after autologous stem cell transplant or as planned tandem (allogeneic transplant after prior autologous stem cell transplant) * Waldenstroms macroglobulinemia (failed one standard regimen) * High grade NHL and diffuse large B-cell lymphoma (DLBCL) * Not eligible for conventional myeloablative HSCT OR failed autologous HSCT * First remission lymphoblastic lymphoma, or small, non-cleaved cell lymphoma or mantle cell lymphoma * Hodgkin disease: * Relapsed or refractory after front-line therapy * Failed or were not eligible for autologous transplantation * Failed prior autotransplant * Age \>= 3 and =\< 75 years for blood and bone marrow transplants and age \>= 3, \< 60 for cord blood transplants * No serious uncontrolled psychiatric illness * No concomitant active malignancy other than non-melanoma skin cancer * Non-pregnant and non-nursing women (women or men with reproductive potential should agree to use an effective means of birth control) * Patients may have received prior autologous bone marrow transplant (BMT) or prior myeloablative allogeneic BMT (at least 60 days have elapsed) * At least 2 weeks since prior chemotherapy, radiation treatment and/or surgery * Informed consent * DONOR: Permissible HLA matching: Related donors- single antigen mismatch at HLA A, B or DRB 1; unrelated donors- a single antigen mismatch at HLA A, B, or C, +/- additional single allele level mismatch at A, B, C or DRB1; cord blood \>= 4 out of 6 antigen match at HLA A, B, DRB1) * DONOR: Compatibility at the four most informative HLA loci: A, B, C and DRB1 are important for reducing the risk of GVHD and successful transplant outcomes; the A, B, C and DRB1 loci comprise 8 possible alleles (a haplotype being inherited from each parent); one additional locus, HLA-DQ, is also typed to ascertain haplotypes and assist in the search for a compatible donor; however mismatching at DQ has not been shown to be associated with adverse outcomes; high resolution molecular typing (at the allele level) is now the standard of care for unrelated donor searches and allows greater refinement of the search strategy * DONOR: Matched related donor: a single antigen mismatch at A, B, or the DR transplant from a family member is associated with a higher risk of GVHD but similar overall survival when compared to full identity at these 3 regions; related donor/recipient pairs must be matched at 5 of 6 HLA antigens (A, B, DRBl) * DONOR: Unrelated donor: when evaluating patients for unrelated donor transplant, a higher degree of matching is preferred due to minimize the risk of GVHD; the A, B, C, DRB1 and DQ loci, comprising 10 possible alleles, will be typed routinely for all unrelated transplants; given the higher risk of TRM in mismatched transplants, RIT is often the best way to mitigate the risk; evolving data from the National Marrow Donor Program now makes it possible to estimate the risks of donor-recipient HLA mismatch at the allele or antigen level; the higher risk from HLA-mismatching must be carefully assessed with respect to the clinical urgency and the patient's risk by the transplant physician; antigen level mismatches at DQ are inconsequential to transplant outcomes and are ignored with respect to donor selection for the purposes of this protocol, with matching requirements confined to the 8 loci involving HLA A, B, C and DRB1; for the purpose of this protocol, a single antigen mismatch at HLA A, B, or C, with or without additional single allele level mismatch may participate in this protocol for voluntary unrelated donors (blood or marrow); patients must be at least antigen-level matched at DRB1 * DONOR: If a patient has no suitable family donor matched for 5 of 6 HLA antigens (A, B, DRB1) and no suitable unrelated donor is identified or for reasons of urgency, the patient can be considered a candidate for cord blood transplant, provided a cord blood donor is identified with a \>= 4 out of 6 antigen match at HLA A, B, DRB1 antigens; the cord blood product must provide a minimum of 2 x 10\^7 nucleated cells/kg, test negative for HIV and Hepatitis A, Band C, and sterility assays have no growth; the cord blood products are located through the National Marrow Donor Program, the American Registry, or the Bone Marrow Donor Worldwide or other established registries, and may be stored in the N.Y Placental Cord Blood Bank, the St. Louis Cord Blood Bank, or any of the established, registered International blood and marrow banks * DONOR: Donor must be healthy and have nonreactive test results for all infectious disease assays as required by state and federal regulations; donors who screen seropositive for hepatitis and/or syphilis must be cleared by infectious disease consultation * DONOR: The donor must have no uncontrolled cardiopulmonary, renal, endocrine, hepatic or psychiatric disease to render donation unsafe * DONOR: The donor must be able to give informed consent for peripheral blood stem cell collection or bone marrow collection * DONOR: Syngeneic donors are not eligible * DONOR: Donors who have poor peripheral venous access, may require central venous line placement for stem cell apheresis Exclusion Criteria: * Uncontrolled central nervous system (CNS) disease (for hematologic malignancies) * Karnofsky (adult) or Lansky (for =\< 16 years) performance status =\< 50% * Diffusing capacity of the lung for carbon monoxide (DLCO) less than 40% predicted, corrected for hemoglobin (Hb) and/or alveolar ventilation * Cardiac: left ventricular ejection fraction less than 40% * Bilirubin \>= 3 x upper limit of normal * Liver alkaline phosphatase \>= 3 x upper limit of normal * Serum glutamic oxaloacetic transaminase (SGOT) or serum glutamic pyruvate transaminase (SGPT) \>= 3 x upper limit of normal * Child's class B and C liver failure * Calculated creatinine clearance \< 40 cc/min by the modified Cockcroft-Gault formula for adults or the Schwartz formula for pediatrics * Patients who have received maximally allowed doses (given in 2 Gy fractions, or equivalent) of previous radiation therapy to various organs as follows: * Mediastinum 40 Gy * Heart (any volume) 36 Gy * Whole lungs 12 Gy * Small bowel (any volume) 46 Gy * Kidneys 12 Gy * Whole liver 20 Gy * Spinal cord (any volume) 36 Gy * Whole brain 30 Gy Enrollment of patients who previously receive higher than allowed dose of radiation to a small volume of lungs, liver, and brain will be determine by the discretion of the radiation oncologist on the study * Uncontrolled diabetes mellitus, cardiovascular disease, active serious infection or other condition which, in the opinion of treating physician, would make this protocol unreasonably hazardous for the patient * Human immunodeficiency virus (HIV) positive * Patients who in the opinion of the treating physician are unlikely to comply with the restrictions of allogeneic stem cell transplantation based on formal psychosocial screening * Females of childbearing potential with a positive pregnancy test