Umbilical Cord Blood Transplant With Added Sugar and Chemotherapy and Radiation Therapy in Treating Patients With Leukemia or Lymphoma

Inclusion Criteria

• •The first 6 patients must be \>= 18 and =\< 65 years old. The subsequent patients may include pediatric patients \>= 12 and =\< 65 years old. Eligibility for pediatric patients will be determined in conjunction with an MD Anderson Cancer Center (MDACC) pediatrician.
• •Performance score of at least 80% by Karnofsky or performance score (PS) \< 3 (Eastern Cooperative Oncology Group \[ECOG\]) (age \>= 12 years)
• •Left ventricular ejection fraction of \> 40%.
• •Pulmonary function test (PFT) demonstrating a diffusion capacity of least 50% predicted.
• •Creatinine =\< 1.5 mg/dL for patients 12 years old and older and =\< 1 for patients younger than 12 years old.
• •Serum glutamate pyruvate transaminase (SGPT) =\< to 2.0 x normal.
• •Bilirubin =\< to 2.0 x normal.
• •Negative beta human chorionic gonadotropin (HCG) test in a woman with child bearing potential defined as not post-menopausal for 12 months or no previous surgical sterilization and willing to use an effective contraceptive measure while on study.
• •Patients must have two cord blood (CB) units available which are matched with the patient at 4, 5, or 6/6 HLA class I (serological) and II (molecular) antigens. Each cord must contain at least 1.5 x 10\^7 total nucleated cells/Kg recipient body weight (pre-thaw).
• •Have identified a backup cells source in case of engraftment failure. The source can be autologous, related or unrelated.
• •Patient must not have a 10/10 HLA matched family member or unrelated donor.
• •Patients will have a back-up graft from any of the following: an available fraction of autologous marrow; or peripheral blood progenitor cells (PBPCs) harvested and cryopreserved; or family member donor; or a third cord blood unit.
• •Prior to initiating chemotherapy in this study, twenty-one or more days must have elapsed since the patient's last radiation or chemotherapy administration (Hydrea, Gleevec and other tyrosine kinase inhibitors \[TKI\] as well as intrathecal therapy are accepted exceptions).