Brief Summary: This phase I/II trial studies the side effects and best dose of gene therapy in treating patients with human immunodeficiency virus (HIV)-related lymphoma that did not respond to therapy or came back after an original response receiving stem cell transplant. In gene therapy, small stretches of deoxyribonucleic acid (DNA) called "anti-HIV genes" are introduced into the stem cells in the laboratory to make the gene therapy product used in this study. The type of anti-HIV genes and therapy in this study may make the patient's immune cells more resistant to HIV-1 and prevent new immune cells from getting infected with HIV-1.

Detailed Description: PRIMARY OBJECTIVE: I. Safety, defined as timely engraftment (the collective establishment of a persistent absolute neutrophil count of at least 500 cells/mm\^3 and platelet count of 20,000 cells/mm\^3 without transfusion for 3 consecutive measurements of laboratory values obtained on different days) by one month post-transplant, in the absence of any grade 3 and 4 non-hematopoietic organ toxicity that can be attributed (possibly, probably, or definitely) to lentiviral transduced stem cell transplant, excluding alopecia, or any clonal expansion and excluding expected toxicities that are associated with the pre-transplant conditioning regimen. SECONDARY OBJECTIVES: I. To determine efficacy of the candidate product, defined as establishment of \> 5% mononuclear blood cells expressing anti-HIV genes in the peripheral blood at 3 months post-transplant. II. To determine the presence, quantity, and duration of gene modified HIV-1 resistant peripheral blood cells and gut mucosal immune cells. III. To study the integration sites of vector sequences in circulating cells. IV. To study progression-free survival. V. To study overall survival. VI. To study complete response rate and duration. VII. To study partial response rate and duration. VIII. To study time to neutrophil engraftment (first measurement of 3 consecutive laboratory values on different days) of absolute neutrophil count \[ANC\] \>= 500 cells/mm\^3). IX. To study time to platelet engraftment (first measurement of 3 consecutive measurements laboratory values obtained on different days) of platelets \>= 20,000 cells/mm\^3 without platelet transfusions 7 days prior). X. To study hematologic function at day 100 (ANC \> 1500, hemoglobin \[Hb\] \> 10 g/dl without transfusion and platelets \> 100,000) XI. To study CD4 recovery at the conclusion of the trial. XII. To study safety in terms of toxicities, infections, transfusions, and infusion-related reactions. XIII. To study HIV-1 viral load over time. XIV. To study persistence of vector-transduced cells over time. EXPLORATORY OBJECTIVE: I. To evaluate the presence and the magnitude of expansion of HIV-1 resistant immune cells in the peripheral blood and gut mucosa of transplanted participants, subsequent to withholding anti-retroviral therapy (ART). OUTLINE: This is a dose-escalation study of lentivirus vector CCR5 shRNA/TRIM5alpha/TAR decoy-transduced autologous CD34-positive hematopoietic progenitor cells. Patients receive BEAM or BEAM-R regimen administered as standard of care comprising carmustine on day -6, cytarabine twice daily (BID) on days -5 to -2, etoposide BID on days -5 to -2, and melphalan on day -1. Patients with B-cell lymphoma also receive rituximab on day -6 before chemotherapy and on days 21 and 28 post-transplant as standard of care. Patients undergo intravenous (IV) infusion of lentivirus vector CCR5 shRNA/TRIM5alpha/TAR decoy-transduced autologous CD34-positive hematopoietic progenitor cells over 1 hour. After completion of study treatment, patients are followed up at days 7, 14, 21, 28, 42, 60, 90, 120, 180, 240, 300, 360, 420, 480, 520, 600, 660, and 720, and then yearly for at least 15 years.