This phase II trial studies the side effects of a cord blood transplant using OTS and to see how well it works in treating patients with human immunodeficiency virus (HIV) positive hematologic (blood) cancers. After a cord blood transplant, the immune cells, including white blood cells, can take a while to recover, putting the patient at increased risk of infection. OTS consists of blood stem cells that help to produce mature blood cells, including immune cells. Drugs used in chemotherapy, such as fludarabine, cyclophosphamide, and thiotepa, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Total body irradiation is a type of whole-body radiation. Giving chemotherapy and total-body irradiation before a cord blood transplant with OTS may help to kill any cancer cells that are in the body and make room in the patient's bone marrow for new stem cells to grow and reduce the risk of infection.