Combination Chemotherapy and Total-Body Irradiation Before Donor Umbilical Cord Blood Transplant in Treating Patients With Advanced Hematologic Cancer, Metastatic Breast Cancer, or Kidney Cancer

DISEASE CHARACTERISTICS: * Diagnosis of 1 of the following: * Acute myeloid leukemia (AML), meeting 1 of the following criteria: * In first complete remission (CR1) by morphology AND at high risk, as evidenced by 1 of the following: * AML secondary to myelodysplastic syndromes (MDS) * High-risk cytogenetics, such as those associated with MDS or complex karyotype * More than 2 courses of therap were required to obtain a CR * In second or greater CR by morphology * In morphologic relapse or persistent disease, defined as \> 5% blasts in normocellular bone marrow OR any percentage of blasts if blasts have unique morphologic markers (e.g., auer rods) * In cytogenetic relapse (without morphologic relapse) * Acute lymphoblastic leukemia (ALL), meeting 1 of the following criteria: * In CR1 by morphology AND at high risk, as evidenced by 1 of the following: * High-risk cytogenetics, such as t(9;22), t(1;19), t(4;11), or other MLL rearrangements * More than 1 course of therapy was required to obtain a CR * In second or greater CR by morphology * In morphologic relapse or persistent disease as defined for AML * In cytogenetic relapse (without morphologic relapse) * Chronic myelogenous leukemia * All types allowed except refractory blast crisis * Non-Hodgkin's lymphoma (NHL) * No intermediate- or high-grade NHL or mantle cell NHL that is progressive on salvage therapy * Stable disease allowed provided it is non-bulky * Hodgkin's lymphoma * No progressive disease on salvage therapy * Stable disease allowed provided it is non-bulky * Chronic lymphocytic leukemia * Multiple myeloma * MDS * Any subtype allowed, including refractory anemia if there is severe pancytopenia or complex cytogenetics * Less than 5% blasts * If patient has blasts ≥ 5% then they must undergo induction therapy before transplantation * Metastatic breast cancer * Disease must have responded to recent chemotherapy OR in plateau after response to chemotherapy * Renal cell cancer * Acquired bone marrow failure syndromes * Small percentage of blasts that is equivocal between marrow regeneration vs early relapse allowed provided there are no associated cytogenetic markers consistent with relapse (for patients with AML or ALL) * Must have a 4/6 HLA-A, -B, and -DRB1 matched unrelated umbilical cord blood donor available * No 5/6 or 6/6 HLA-A , -B, and -DRB1 matched sibling donor available * No more than 2 antigen mismatches at the HLA-A, -B, or -DRB1 loci NOTE: A new classification scheme for adult non-Hodgkin's lymphoma has been adopted by PDQ. The terminology of "indolent" or "aggressive" lymphoma will replace the former terminology of "low", "intermediate", or "high" grade lymphoma. However, this protocol uses the former terminology. PATIENT CHARACTERISTICS: * Karnofsky performance status 60-100% OR Lansky performance status 50-100% (pediatric patients) * Albumin \> 2.5 g/dL * Creatinine ≤ 2.0 mg/dL (adults) OR creatinine clearance \> 40 mL/min (pediatric patients) * Adults with creatinine \> 1.2 mg/dL or a history of renal dysfunction must have a creatinine clearance \> 40 mL/min * Transaminases \< 5 times upper limit of normal (ULN) * Bilirubin \< 3 times ULN * LVEF ≥ 35% * DLCO \> 30% of predicted * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective contraception * No evidence of HIV infection or known HIV-positivity * No decompensated congestive heart failure * No uncontrolled cardiac arrhythmia * No requirement for supplemental oxygen * No active, serious infection * Recent mold infection (e.g., Aspergillus) allowed provided patient has received ≥ 30 days of appropriate treatment AND infection is controlled and cleared by an infectious disease specialist PRIOR CONCURRENT THERAPY: * No prior irradiation that precludes the safe administration of 1 additional dose of 200 cGy of total-body irradiation * At least 3 months since prior myeloablative bone marrow transplantation