A Two-Step Approach to Reduced Intensity Bone Marrow Transplant for Patients With Hematological Malignancies

Inclusion Criteria: 1. Any patient with hematologic or oncologic diagnosis in which allogeneic HSCT is thought to be beneficial, and in whom front-line therapy has already been applied. Patients treated on this protocol will be without morphological evidence of disease (complete remission or "CR"), or if the patient has evidence of disease, the patient must have had at least a good partial response (PR) to the most recent therapy and the disease must be chemoresponsive. 2. Patients treated on this study will have: * Acute leukemia in 1st or 2nd CR * MDS (myelodysplastic syndrome), specific subtypes of RA (refractory anemia) or RARS (refractory anemia with ringed sideroblasts) subtypes. * Hodgkin or Indolent Non-Hodgkin's lymphoma with chemosensitive disease * Myeloma without morphological evidence of disease, or a deep PR to the most recent therapy * Myeloproliferative disorders with at least a PR to current therapy * Aplastic Anemia * A hematological or oncological disease (not listed) that meets the criteria reviewed above (in CR or with a good PR). 3. Patients must have a related donor who is HLA mismatched at 2, 3, or 4 antigens at the HLA-A; B; C; DR loci in the GVHD direction. (Patients with related donors who are HLA identical or are a 1-antigen mismatch may be treated on this therapeutic approach, but will have their outcomes will not be part of the statistical aims of the study (see Summary section). 4. Patients must adequate organ function: * LVEF (Left ventricular end diastolic function) of \>50% * DLCO (Diffusing Capacity of the Lung for Carbon Monoxide ) ≥50% of predicted corrected for hemoglobin * Adequate liver function as defined by a serum bilirubin \<1.8, AST or ALT \< 2.5X upper limit of normal * Creatinine Clearance of ≥ 60 mL/min 5. Performance status ≥ 80% (TJU Karnofsky) for patients ≥ 60 years old or ≥70% for patients \< 60 years old. 6. HCT-CI Score ≤ 4 points for patients ≥ 60 years old or ≤ 5 points for patients \< 60 years old. • Patients must have adequate Karnofsky performance status (KPS) and hematopoietic cell transplantation comorbidity index (HCT-CI) scores: * Patients \< age 60 years must have a KPS of \>= 80% and an HCT-CI score of 5 or less * Patients aged 60 to 65 years must have a KPS of \>= 80% and an HCT-CI score of 4 or less * Patients aged 66 to 69 years must have a KPS of 90% and an HCT-CI score of 3 or less * Patients aged 70 years or more must have a KPS of 90% and an HCT-CI score of 2 or less * Patients with greater than the allowable HCT-CI points for age can be enrolled for trial with approval of the principal investigator (PI) and at least 1 co-investigator (Co-I) not on the primary care team of the patient; this is an adjustment to account for healthy patients who meet the spirit of this protocol but have histories that result in higher than guideline HCT-CI points; an example is a patient with a solid tumor malignancy in their remote history (adds 3 points to HCT-CI total) where the treatment for the malignancy occurred years to decades before and there has been complete recovery of toxicities 7. Patients must be willing to use contraception if they have childbearing potential 8. Able to give informed consent Exclusion Criteria: * Human immunodeficiency virus (HIV) positive * Active involvement of the central nervous system with malignancy * Inability to obtain informed consent * Pregnancy * Patients with life expectancy of =\< 6 months for reasons other than their underlying hematologic/oncologic disorder * Patients who have received alemtuzumab or anti-thymocyte globulin (ATG) within 8 weeks of the transplant admission; (documented by the absence of these agents in the medical record) * Patients with evidence of another malignancy, exclusive of a skin cancer that requires only local treatment, should not be enrolled on this protocol