Gemcitabine, Docetaxel, and Hydroxychloroquine in Treating Participants With Recurrent or Refractory Osteosarcoma

Inclusion Criteria: * Patient or patient's legally acceptable representative has provided informed consent * Patients must have a pathological diagnosis of osteosarcoma at original diagnosis or time of recurrence * Patients must have relapsed or become refractory to conventional therapy, with a regimen containing some combination of high dose methotrexate, doxorubicin, cisplatin, and/or ifosfamide * Patients may have either unresectable or potentially resectable disease. However, patients with potentially resectable disease must be willing to await surgery until after the primary efficacy endpoint is reached at 4 months * Patients must have measurable disease by RECIST 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2. Use Karnofsky for patients \> 16 years old and Lansky for patients =\< 16 * Has archived tumor tissue and is willing to provide a tumor block or unstained slides or the patient is willing to undergo a pretreatment biopsy. Biopsy during cycle #2 of therapy is required. Biopsy at progression is strongly encouraged but will be optional * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L * Platelet count \>= 100 x 10\^9/L * Hemoglobin \> 8 g/dL * Prothrombin time (PT) or partial thromboplastin time (PTT) \< 1.5 x upper limit of normal (ULN), except subjects on anticoagulation for venous thromboembolism * Calculated creatinine clearance or radioisotope glomerular filtration rate (GFR) \> 60 mL/min * Aspartate aminotransferase (AST) and ala…