Umbilical Cord Blood (UCB) Transplant, Fludarabine, Melphalan, and Anti-thymocyte Globulin (ATG) in Treating Patients With Hematologic Cancer

DISEASE CHARACTERISTICS: * Diagnosis of hematologic malignancy for which a reduced-intensity allogeneic stem cell transplantation is deemed clinically appropriate, including any of the following: * Chronic myelogenous leukemia, meeting one of the following criteria: * In first chronic phase AND failed imatinib mesylate therapy, defined as failure to obtain a hematologic remission by 3 months or major cytogenetic response (Ph+ cells \< 35%) by 12 months, or demonstrated clonal evolution or disease progression while on therapy * In accelerated phase with \< 15% blasts * In blast crisis that has entered into a second chronic phase following induction chemotherapy * Acute myelogenous leukemia, meeting one of the following criteria: * In second or subsequent completion remission\* * Failed primary induction chemotherapy, but subsequently entered into a complete remission\* with ≤ 2 subsequent re-induction chemotherapy treatment(s) * In first complete remission\* with poor-risk cytogenetics NOTE: \*Complete remission is defined as \< 5% blasts in bone marrow, no definitive evidence of disease by morphology, flow cytometry, or genetic studies, and no circulating blasts. Neutrophil and platelet count recovery will not be required. * Acute lymphoblastic leukemia, meeting one of the following criteria: * In second or subsequent complete remission * In first complete remission AND t(9;22) * Myelodysplastic syndromes, meeting the following criteria: * High-risk disease, defined as International Prognostic Scoring System score of ≥ 1.5 * Less than 10% blasts at the time of study enrollment * Chronic myelomonocytic leukemia * Less than 10% blasts at the time of study enrollment * Myeloid metaplasia with myelofibrosis with poor-risk features, meeting one of the following criteria: * Age \< 55 years AND a Lille score of 1 * Lille score of 2 * Hemoglobin \< 10 g/dL AND abnormal karyotype * Chronic lymphocytic leukemia/prolymphocytic leukemia, meeting all of the following criteria: * Rai stage I-IV disease * Failed ≥ 1 prior chemotherapy regimen, including fludarabine, or autologous stem cell transplantation * Chemosensitive or stable, non-bulky disease prior to transplant * Received ≤ 3 prior chemotherapy regimens (monoclonal antibody therapy and involved-field radiotherapy are not considered prior regimens) * Low-grade B-cell non-Hodgkin lymphoma (NHL) (small lymphocytic lymphoma, follicular center \[grade 1 or 2\] lymphoma, or marginal zone lymphoma), meeting all of the following criteria: * Failed ≥ 1 prior chemotherapy regimen or autologous stem cell transplantation * Chemosensitive or stable, non-bulky disease prior to transplant * Received ≤ 3 prior chemotherapy regimens (monoclonal antibody therapy and involved-field radiotherapy are not considered prior regimens) * Intermediate-grade B-cell or T-cell NHL or mantle cell NHL, meeting all of the following criteria: * Failed to achieve remission or recurred after either conventional chemotherapy or autologous stem cell transplantation * Chemosensitive, non-bulky disease prior to transplant * Hodgkin lymphoma, meeting all of the following criteria: * Relapsed after prior autologous stem cell transplantation or after ≥ 2 combination chemotherapy regimens AND ineligible for autologous peripheral blood stem cell transplantation * Chemosensitive, non-bulky disease prior to transplant * Multiple myeloma, meeting one of the following criteria: * Relapsed after autologous stem cell transplantation * Relapsed after conventional therapies AND not a candidate for autologous stem cell transplantation * No HLA-matched related or unrelated donor available * Has two umbilical cord blood units available that are matched at ≥ 4/6 HLA A, B, and DRB1 with the patient and with each other (HLA C and DQ will not be used in the match strategy) * Total combined nucleated cell dose from the 2 umbilical cord blood units must be \> 3.7 x 10\^7 nucleated cells/kg (pre-freeze dose) 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 80-100% * Adapted, weighted Charlson Comorbidity Index \< 3 * Serum creatinine ≤ 2.0 mg/dL * AST or ALT \< 3 times upper limit of normal (ULN) * Bilirubin \< 1.5 times ULN * Not pregnant or nursing * LVEF ≥ 40% * DLCO \> 50% * No hypoxia at rest with oxygen saturation \< 92% on room air (corrected with bronchodilator therapy) * No active opportunistic infection (e.g., fungal pneumonia, tuberculosis, or viral infection) * No active hepatitis B or C infection that, in the opinion of a gastroenterologist or the transplant committee, places the patient at moderate- to high-risk for developing severe hepatic disease * No HIV infection PRIOR CONCURRENT THERAPY: * See Disease Characteristics