Inclusion Criteria:  Radiation-based cohort diagnoses:  Acute myeloid leukemia Acute lymphoid leukemia in remission Myelodysplasia (MDS) Chronic lymphocytic leukemia (CLL) with no or minimal lymph node involvement Multiple myeloma Chronic myeloid leukemia Myelofibrosis Myeloid malignancy not otherwise specified Chronic myelomonocytic leukemia Essential thrombocytopenia or polycythemia vera T cell leukemia T cell lymphoma without significant lymph node disease burden Any hematological malignancy or dyscrasia not cited above in which HSCT is potentially curable * Any patient who has a hematological disease that would normally be treated on a myeloablative study, but is prevented from doing so by factors in their past medical history. Examples are patients with previous treatment with radiation therapy precluding total-body irradiation (TBI), or a past history of myeloablative therapy, precluding a 2nd myeloablative regimen. Patients must have a donor who is one-haplotype mismatched (number of mismatches in either direction not considered)  Chemotherapy-based cohort diagnoses:  Hodgkin or non-Hodgkin lymphoma Small lymphocytic lymphoma/CLL Any other diagnosis in which chemotherapy is thought to be superior to radiotherapy for treatment of the disease Hematological malignancy in patients who cannot receive > 2 Gy radiation * Aplastic anemia and other non-malignant hematologic dyscrasias Patients must have a donor who is one-haplotype mismatched (number of mismatches in either direction not considered)  HLA identical cohort diagnoses:  * Patients in this group will be treated in parallel to the radiation-based cohort or the chemotherapy-based group based on what category their diagnosis falls into. However, these patients will have HLA identical related donors (one-antigen cross-over event included).  Left ventricular ejection fraction of >= 50% Diffusion lung capacity of oxygen >= 50% and forced expiratory volume at 1 second >= 50% of predicted corrected for hemoglobin Adequate liver function as defined by a serum bilirubin =< 1.8, aspartate aminotransferase or alanine aminotransferase =< 2.5 x upper limit of normal Creatinine clearance of >= 60 mL/min  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 and at least 1 co-investigator 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 Patients must be willing to use contraception if they have childbearing potential Patient or patient's guardian is able to give informed consent Patients should have a life expectancy of >= 6 months for reasons other than their underlying hematologic/oncologic disorder  Exclusion Criteria:  Patients with evidence of another malignancy, exclusive of a skin cancer that requires only local treatment, should not be enrolled on this protocol  Patients should not be:  Human immunodeficiency virus positive Have active involvement of the central nervous system with malignancy. This can be documented by a normal neurological exam, magnetic resonance imaging (MRI) of the head, and/or a negative cerebral spinal fluid analysis Pregnant or breastfeeding