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