Fludarabine Phosphate, Melphalan, and Low-Dose Total-Body Irradiation Followed by Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Hematologic Malignancies

Inclusion Criteria: * Diagnosis of a histology documented hematologic malignancy or marrow disorder BONE MARROW FAILURE DISORDERS: * Acquired bone marrow failure disorders include aplastic anemia, paroxysmal nocturnal hemoglobinuria (PNH) \* Primary allogeneic 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 available \* Patients with PNH should not be eligible for a myeloablative HSCT * Hereditary bone marrow failure disorders include Diamond-Blackfan Anemia, Shwachman-Diamond Syndrome, Kostmann Syndrome, congenital Amegakaryocytic Thrombocytopenia; Fanconi Anemia or related chromosomal breakage syndrome, Dyskeratosis Congenital are excluded from this study die to their poor deoxyribonucleic acid (DNA) repair capacity \* Fanconi anemia or related chromosomal breakage syndrome: positive chromosome breakage analysis using diepoxybutane (DEB) or mitomycin C if applicable \* Dyskeratosis Congenita: diagnosis is supported by using either telomerase RNA component (TERC) gene mutation in autosomal dominant Dyskeratosis Congenita or X-linked 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 immunodeficiency syndrome, Wiskott-Aldrich syndrome, CD40 ligand deficiency, T-cell deficiencies) ACUTE LEUKEMIAS: * Subjects must be ineligible for or unable to receive a conventional myeloablative transplantation * Resistant or recurrent disease after at least one standard combination chemotherapy 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 mutations (e.g., fms-like tyrosine kinase3-internal tandem duplication \[Flt3-ITD\] mutation) \* 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 SYNDROME (MDS): * Myelofibrosis (with/without splenectomy) with intermediate to high risk features * Advanced polycythemia vera nor responding to standard therapy * MDS with lower International Prognostic Scoring System (IPSS) score of intermediate (Int)-2 or higher * MDS with lower IPSS score Int-1 or less with severe clinical features such as severe neutropenia or thrombocytopenia or high risk chromosome abnormalities such as monosomy 7 * Secondary MDS with any IPSS scores * Chronic myelomonocytic 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 or complete remission (CR) between courses of conventional therapy * Multiple myeloma (progressive disease after autologous stem cell transplant, tandem allogeneic transplant after prior autologous stem cell transplant) * Waldenstrom's 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 LYMPHOMA: * Received and failed front-line therapy * Failed or were not eligible for autologous transplantation DONOR: Permissible human leukocyte antigen (HLA) matching: related donors * single antigen mismatch at HLA A, B, or DRB1; unrelated donors * a single antigen mismatch at HLA A, B, or C, +/- additional single allele level mismatch at A, B, V or DRB1 * Minimum goal for peripheral blood stem cells (PBSC) dose is 2 x 10\^6 CD34+ cells/kg of recipient weight; minimum goal for the marrow dose is 1 x 10\^8 nucleated cells/kg of recipient weight * No serious uncontrolled psychiatric illness * No concomitant active malignancy that would be expected to require chemotherapy within 3 years of transplant (other than non-melanoma skin cancer) * Non-pregnant and non-nursing woman; (women or men with reproductive potential should agree to use an effective means of birth control) * Patients who have failed a prior autologous or allogeneic transplant are eligible; however, at least 90 days must have elapsed between the start of this reduced intensity conditioning regimen and the last transplant if patient had a prior autologous or myeloablative allogeneic bone marrow transplant (BMT) * At least 2 weeks since prior chemotherapy, radiation treatment and/or surgery * Informed consent 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, DRB1) DONOR: Unrelated Donor: When evaluating patients for unrelated donor transplant, the higher degree of matching, the lower risk of GvHD; the A, B, C, DRB1 and DQB1 loci, comprising 10 possible antigen (with alleles), will be typed for all unrelated transplants; given the higher risk of TRM in mismatched transplants, RIT is often the best way to mitigate the risk; data from the National Marrow Donor Program makes it possible to estimate the risk of donor-recipient HLA mismatch at the allele or antigen level; the higher risk from HLA-mismatching must be balanced against the clinical urgency and the patient's risk by the transplant team; at this time, antigen level mismatches at DQB1 do not affect outcomes and will not be used for matching purposes for donor selection; thus, the matching required will be at the HLA A, B, C and DRB1 (8 loci); for this protocol, a single antigen mismatch at the HLA A, B, C, with or without additional single allele level mismatch may participate in this protocol for voluntary unrelated donors (blood or marrow) DONOR: Donor must be healthy and have non-reactive test results for all infectious disease assays as required by state and federal regulations; donors who screen seropositive for hepatitis an/or syphilis must be cleared by infectious disease consultation DONOR: Donor must have no uncontrolled cardiopulmonary, renal, endocrine, hepatic or psychiatric disease to render donation unsafe DONOR: The donor (or parent in minor) must 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) \< 40% predicted, corrected for hemoglobin and/or alveolar ventilation * Left ventricular ejection fraction \< 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 Cockroft-Gault formula for adults or the Schwartz formula for pediatrics * Patients who have received maximally allowed doses (given in 2 Gy fractionations, or equivalent) of previous radiation therapy to various organs as follows: \* Mediastinum: adult -40, pediatric (=\<18 yrs) - 21 \* Heart: adult 36, pediatric - 26 \* Whole lung(s): adult - 12, pediatric - 10 \* Small bowel: adult - 46, pediatric - 40 \* Kidneys: adult - 12, pediatric - 10 \* Whole liver: adult - 20, pediatric - 20 \* Spinal cord: adult - 36, pediatric - 36 \* Whole Brain: adult 30, pediatric - 30 * Patients who previously have received a higher than allowed dose of radiation to a small lung, liver, and brain volume, will be evaluated by the radiation oncologist to determine if the patient is eligible for 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 * Female of childbearing potential with a positive pregnancy test