Total-Body Irradiation and Fludarabine Phosphate Followed by Donor Peripheral Blood Stem Cell Transplant in Treating Patients With Hematologic Malignancies or Kidney Cancer

Inclusion Criteria: * Ages \> 50 years with hematologic malignancies treatable by unrelated hematopoietic stem cell transplantation (HSCT) * Ages =\< 50 years of age with hematologic diseases treatable by allogeneic HSCT who through pre-existing medical conditions or prior therapy are considered to be at high risk for regimen related toxicity associated with a conventional transplant (\> 40% risk of transplant related mortality \[TRM\]) or those patients who refuse a conventional HSCT; 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 at the collaborating center; patients =\< 50 years of age who have received previous autologous transplantation do not require patient review committee approval; all children \< 12 years must be discussed with the FHCRC principal investigator (PI) prior to registration * 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: * Intermediate or high grade non-Hodgkin lymphoma (NHL) - not eligible for autologous HSCT or after failed autologous HSCT * Low grade NHL - with \< 6 month duration of complete remission (CR) between courses of conventional therapy * Chronic lymphocytic leukemia (CLL) - must have failed two lines of conventional therapy and be refractory to fludarabine * Hodgkin's disease (HD) - must have received and failed frontline therapy * Multiple myeloma (MM) - must have received prior chemotherapy; consolidation of chemotherapy by autografting prior to nonmyeloablative HSCT is permitted * Acute myeloid leukemia (AML) - must have \< 5% marrow blasts at the time of transplant * Acute lymphoblastic leukemia - must have \< 5% blasts at the time of transplant * 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 with PBSC autologous or conventional HSCT for advanced CML may be enrolled provided they are in CR or chronic phase (CP) and have \< 5% marrow blasts at time of transplant * Myelodysplastic syndromes (MDS)/myeloproliferative disorder (MPD) - only patients with MDS/refractory anemia (RA) or MDS/refractory anemia with ringed sideroblasts (RARS) will be eligible for this protocol; additionally patients with myeloproliferative syndromes (MPS) will be eligible; those patients with MDS or MPS with \> 5% marrow blasts (including those with transformation to AML) must receive cytotoxic chemotherapy and achieve \< 5% marrow blasts at time of transplant * Renal cell carcinoma - must have evidence of disease not amenable to surgical cure or history of or active metastatic disease by radiological and histologic criteria * DONOR: FHCRC matching allowed will be grade 1.0 to 2.1; unrelated donors who are prospectively: * Matched for human leukocyte antigen (HLA)-A, B, C, DRB1 and DQB1 by high resolution typing * Only a single allele disparity will be allowed for HLA-A, B, or C as defined by high resolution typing * DONOR: A positive anti-donor cytotoxic crossmatch is an absolute donor exclusion * DONOR: Patient and donor pairs homozygous at a mismatched allele are considered a two-allele mismatch, i.e., the patient is A\*0101 and the donor is A\*0201, and this type of mismatch is not allowed * DONOR: PBSC only will be permitted as a hematopoietic stem cell (HSC) source on this protocol Exclusion Criteria: * Patients with rapidly progressive intermediate or high grade NHL * Renal cell carcinoma patients: * With expected survival of less than 6 months * Disease resulting in severely limited performance status (\< 70%) * Any vertebral instability * History of brain metastases * Central nervous system (CNS) involvement with disease refractory to intrathecal chemotherapy * 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 except renal cell carcinoma * Fungal infections with radiological progression after receipt of amphotericin B or active triazole for greater than 1 month * Cardiac ejection fraction \< 35%; ejection fraction is required if there is a history of anthracycline exposure or history of cardiac disease * Diffusion capacity of the lung for carbon monoxide (DLCO) \< 40% and/or receiving supplementary continuous oxygen * The FHCRC PI of the study must approve of enrollment of all patients with pulmonary nodules * Patients with clinical or laboratory evidence of liver disease would be evaluated for the cause of liver disease, its clinical severity in terms of liver function, and the degree of portal hypertension; patients will be excluded if they are found to have fulminant liver failure, cirrhosis of the liver with evidence of portal hypertension, alcoholic hepatitis, esophageal varices, a history of bleeding esophageal varices, hepatic encephalopathy, uncorrectable hepatic synthetic dysfunction evinced by prolongation of the prothrombin time, ascites related to portal hypertension, bacterial or fungal liver abscess, biliary obstruction, chronic viral hepatitis with total serum bilirubin \> 3 mg/dL, and symptomatic biliary disease * Karnofsky scores \< 60 (except renal cell carcinoma \[RCC\]) * Patients with \> grade II hypertension by common toxicity criteria (CTC) * Human immunodeficiency virus (HIV) positive patients * The addition of cytotoxic agents for "cytoreduction" with the exception of hydroxyurea and imatinib mesylate will not be allowed within two weeks of the initiation of conditioning * DONOR: Marrow donors * DONOR: Donors who are HIV-positive and/or, medical conditions that would result in increased risk for filgrastim (G-CSF) mobilization and harvest of PBSC