This phase I clinical trial evaluates the safety and feasibility of administering NY-ESO-1 TCR (T cell receptor)engineered peripheral blood mononuclear cells (PBMC) and peripheral blood stem cells (PBSC) after a myeloablative conditioning regimen to treat patients with cancer that has spread to other parts of the body. The conditioning chemotherapy makes room in the patient?s bone marrow for new blood cells (PBMC) and blood-forming cells (stem cells) to grow. Giving NY-ESO-1 TCR PBMC and stem cells after the conditioning chemotherapy is intended to replace the immune system with new immune cells that have been redirected to attack and kill the cancer cells and thereby improve immune system function against cancer.