Fludarabine Phosphate, Low-Dose Total Body Irradiation, and Donor Stem Cell Transplant in Treating Patients With Hematologic Malignancies or Kidney Cancer

Inclusion Criteria: * Age \> 50 years with hematologic malignancies treatable by allogeneic hematopoietic stem cell transplant (HSCT) and all patients with B cell malignancies except those who may be cured by autologous stem cell transplantation (SCT) * Age =\< 50 years of age with hematologic diseases treatable by allogeneic HSCT who through pre-existing medical conditions or prior therapy are considered to be of high risk for regimen related toxicity associated with a conventional transplant or those patients who refuse a conventional SCT; transplants must be approved for these inclusion criteria by both the participating institution's patient review committee such as the Patient Care Conference (PCC at the Fred Hutchinson Cancer Research Center \[FHCRC\]) and by the principal investigator * Patients with metastatic renal cell carcinoma with the histologic subtypes of clear cell, papillary and medullary may be accepted regardless of age * The following diseases will be permitted although other diagnoses can be considered if approved by PCC or the participating institution's patient review committees and the principal investigator * Non-Hodgkin lymphoma (NHL), chronic lymphocytic leukemia (CLL), Hodgkin lymphoma (HL) - must have received and failed frontline therapy * Multiple myeloma - must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative HSCT is permitted * Acute myeloid leukemia (AML)/acute lymphoblastic leukemia (ALL) - must be in complete remission and have received cytotoxic chemotherapy at some stage before transplant; patients with molecular or early relapse will be accepted providing a donor is available; patients with persistent or refractory disease will be considered on a case by case basis and transplants must be approved by the institution's patient review committees * Chronic myelogenous leukemia (CML) - patients will be accepted in chronic phase or accelerated phase; patients who have received prior autografts after high dose therapy or have undergone intensive chemotherapy for either peripheral blood stem cell (PBSC) mobilization or treatment of advanced CML may be enrolled provided they are in complete remission (CR), chronic phase (CP) or accelerated phase (AP) * Myelodysplastic syndromes (MDS) - all patients with MDS will be eligible for this protocol; however, those patients with MDS and frank AML (\> 30% blasts in bone marrow aspirate by morphology or flow cytometry) will require induction chemotherapy to obtain a complete remission (marrow blasts \< 5%) and remain in complete remission at time of transplant * Renal cell carcinoma- must have evidence of disease not amenable to surgical cure or metastatic disease by radiological and histological criteria * DONOR: Human leukocyte antigen (HLA) matched unrelated donor; donors should be matched for HLA -A, -B, -C, -developmentally regulated ribonucleic acid (RNA) binding protein 1(DRB)1 and -class II, DQ beta 1 (DQB) 1; HLA -A and -B loci should be matched at least to the level of resolution; HLA -C, -DRB1, and -DQB1 should be typed at the highest level of resolution available at the time of donor selection; donor must consent to either a bone marrow harvest or PBSC mobilization with filgrastim (G-CSF) arranged through the National Marrow Donor Program (NMDP) or other donor centers Exclusion Criteria: * Patients with rapidly progressive intermediate or high grade NHL * Renal cell carcinoma patients with expected survival of less than 6 months * Bulky disease resulting in severely limited performance status (\< 70%) * Any vertebral instability * Any active central nervous system (CNS) involvement with disease * Fertile men or women unwilling to use contraceptive techniques during and for 12 months following treatment * Females who are pregnant * Patients with non-hematological tumors * Cardiac ejection fraction \< 30% * Diffusion capacity of the lung for carbon monoxide (DLCO) \< 30% and/or receiving supplementary continuous oxygen * Significant elevation of bilirubin and transaminases should be discussed at participating institutions' patient review committees in a case by case basis; evidence of synthetic dysfunction or severe cirrhosis will result in patient exclusion * Karnofsky score \< 50 (except renal cell carcinoma \[RCC\]) * Patients with poorly controlled hypertension on multiple antihypertensives * Human immunodeficiency virus (HIV) positive patients