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 g…