DISEASE CHARACTERISTICS: Histologically confirmed malignancy including: - Pediatric acute lymphocytic leukemia (ALL) in first complete remission with high risk features including presence of t(4;11) or t(9;22), or extreme hyperleukocytosis (initial WBC greater than 500 K/mL), or failure to achieve a complete remission after standard induction therapy - Adult ALL or acute nonlymphoblastic leukemia (ANLL) in first complete remission with t(8;14) translocation or failure to achieve complete remission after standard induction therapy - ALL or ANLL in second or subsequent remission - Chronic myelogenous leukemia in chronic or accelerated phase - Myelodysplastic syndrome with evidence of evolution to acute myeloid leukemia Refractory anemia with excess blasts Refractory anemia with excess blasts in transformation Chronic myelomonocytic leukemia - T-lymphoblastic non-Hodgkin's lymphoma in second or subsequent remission - Stage IV neuroblastoma Must also meet all the following conditions: No HLA-ABC/DR identical related bone marrow or UCB donor No 5/6 antigen matched related bone marrow or UCB donor Condition precludes waiting to search and find a donor in the National Marrow Donor Registry Must have an available serologic matched umbilical cord blood unit in the New York Blood Center's Placental Blood Project No active CNS disease Not eligible for COBLT study (Transplantation of Banked Umbilicial Cord Blood Cells for Use in Clinical Research on Transplantation of Umbilical Cord Blood Stem and Progenitor Cells)