DISEASE CHARACTERISTICS: Histologically proven hematologic malignancy Acute lymphocytic leukemia (ALL): In second or later remission In first remission with poor prognostic features (Philadelphia chromosome positive) Acute myeloid leukemia (AML): In second or later remission In first remission with poor prognostic features, e.g., Arising from myelodysplastic syndrome Cytogenetics with -5, -7, +8, 11q23 abnormalities Complex cytogenetics AML or ALL refractory to induction or in relapse Myelodysplastic syndrome Chronic myelogenous leukemia Severe aplastic anemia or Fanconi's anemia Relapsed Hodgkin's disease Relapsed non-Hodgkin's lymphoma Multiple myeloma No suitable family donor matched for 5 or 6 HLA antigens (A, B, DR) No suitable unrelated donor matched for 6 HLA antigens Cord blood donor available matched for 4-6 out of 6 HLA antigens

PATIENT CHARACTERISTICS: Age: 5 to 50 Performance status: Karnofsky 70-100% Life expectancy: Not specified Hematopoietic: Not specified Hepatic: Bilirubin less than 3 times normal Alkaline phosphatase less than 3 times normal SGOT less than 3 times normal Renal: Creatinine less than 2 times normal OR Creatinine clearance greater than 60 mL/min Cardiovascular: MUGA with ejection fraction at least 50% Pulmonary: DLCO and spirometry at least 60% OR Exercise VO2 max/kg at least 15 mL/min/kg Other: HIV antibody negative Hepatitis B surface antigen negative No active bacterial, viral, or fungal infection

PRIOR CONCURRENT THERAPY: Biologic therapy: Not specified Chemotherapy: Not specified Endocrine therapy: Not specified Radiotherapy: Prior radiotherapy allowed If following limits have not been exceeded, patient may receive total body irradiation: No prior radiation to one entire kidney Whole liver received no greater than 1000 cGy No prior whole abdomen radiotherapy Small bowel received no greater than 3000 cGy Heart received no greater than 1800 cGy No prior whole lung radiotherapy CNS received less than 30 cGy (whole brain or any portion of the spine) Surgery: Not specified