Umbilical Cord Blood Stem Cell Transplant in Treating Patients With Hematologic Cancer or Other Disease

DISEASE CHARACTERISTICS: * Histologically confirmed diagnosis of 1 of the following: * Acute myeloid leukemia meeting the following criteria: * M0-M7 histologic subtypes by French-American-British classification * Previously treated disease * Meets 1 of the following criteria: * Persistent disease as evidenced by 5-30% persistent blasts in bone marrow after induction or salvage therapy * In second or subsequent complete remission (CR) * In first CR with 1 of the following high-risk features: * Philadelphia chromosome present * Noncore-binding factor type of chromosomal abnormalities * Myelodysplastic syndromes with 1 of the following International Prognostic Scoring System (IPSS) scores: * Intermediate-1 * Intermediate-2 * High-risk score with transfusion dependence * Chronic myelogenous leukemia meeting 1 of the following criteria: * In accelerated or blastic phase * Failed prior imatinib mesylate therapy * Acute lymphoblastic leukemia meeting 1 of the following criteria: * In first CR with any of the following high-risk features: * Philadelphia chromosome present * Translocation t(4;11) present * WBC \> 30,000/mm³ (adult patients) * More than 4 weeks from initiation of treatment was required to achieve CR (adult patients) * DNA index of near haploid (N=23 chromosomes) (pediatric patients) * In second or subsequent CR * Persistent disease as evidenced by 5-20% persistent blasts in bone marrow after induction or salvage therapy * Hodgkin's or non-Hodgkin's lymphoma meeting the following criteria: * Recurrent or refractory disease * Tumor ≤ 5 cm in diameter * Myeloma or plasma cell neoplasm meeting 1 of the following staging criteria: * Stage III at presentation * Stage I-II at presentation * Not responding OR progressed after first-line therapy * Chronic lymphocytic leukemia or Waldenstrom's macroglobulinemia with refractory or progressive disease after first-line therapy * No 5-6/6 HLA-matched related or 7-8/8 HLA-matched unrelated marrow or peripheral blood stem cell donor available * No single 4-6/6 HLA-A, -B, or -DRB1-matched umbilical cord blood unit ≥ 3.5 x 10\^7 nucleated cells/kg available PATIENT CHARACTERISTICS: * ECOG performance status (PS) 0-2 OR Karnofsky or Lansky PS 70-100% * Not pregnant * Fertile patients must use effective contraception prior to and during study participation * HIV negative * Bilirubin \< 3.0 mg/dL * AST and ALT ≤ 3 times upper limit of normal * Creatinine \< 2.0 mg/dL OR creatinine clearance \> 50 mL/min * Cardiac ejection fraction \> 50% by echocardiogram OR shortening fraction \> 27% * No uncontrolled symptomatic congestive heart failure * No unstable angina pectoris * No cardiac arrhythmia * FEV\_1 \> 50% of normal * Forced vital capacity \> 50% of normal * DLCO normal * Oxygen saturation \> 92% on room air (for patients \< 5 years of age) * No history of allergic reactions attributed to compounds of similar chemical or biological composition to busulfan and fludarabine phosphate * No ongoing or active infection * No psychiatric illness or social situation that would preclude study compliance * No other uncontrolled illness PRIOR CONCURRENT THERAPY: * See Disease Characteristics * At least 4 weeks since prior and no concurrent surgery * At least 4 weeks since prior and no other concurrent investigational or commercial agents or therapies for the malignancy, including chemotherapy, biologic therapy, or radiotherapy