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 alanine aminotransferase (ALT) \< 2.5 x ULN for age
* Total bilirubin \< 1.5 x ULN for age
* Serum albumin \>= 2

Exclusion Criteria:

* Any disorder that compromises the ability of the patient or patient's legally acceptable representative to give written informed consent and/or to comply with the study procedures
* Patients who will not be available for protocol required study visits, to the best of the subject and investigator's knowledge
* History of hypersensitivity to any of the study agents (gemcitabine, docetaxel, or hydroxychloroquine) or to any component of the formulation
* Patients who have previously received gemcitabine or docetaxel
* Men and woman of reproductive potential, unwilling to practice a highly effective method of birth control for the duration of the study and continuing for 2 weeks (for women) and 12 weeks (for men) after receiving the last dose of study drug. Highly effective methods of birth control include sexual abstinence (men, women); vasectomy or a condom used with spermicide (men) in combination with barrier methods, hormonal birth control or intrauterine device (IUD) (women)
* Women who are lactating/breastfeeding or planning to become pregnant during the duration of the study
* History or current diagnosis of a second neoplasm. Exception: adequately treated non-melanoma skin cancer, curatively treated in-situ disease, or other solid tumors curatively treated with no evidence of disease for \>= 2 years
* Myocardial infarction within 6 months of study day 1, symptomatic congestive heart failure (New York Heart Association \> class II), unstable angina, or unstable cardiac arrhythmia requiring medication, or uncontrolled hypertension (systolic \> 160 mmHg or diastolic \> 100 mmHg)
* Active infection requiring intravenous (IV) antibiotics within 2 weeks of study enrollment
* Significant gastrointestinal disorder(s) that in the opinion of the investigator (e.g., Crohn's disease, ulcerative colitis, extensive gastrointestinal resection) may influence drug absorption
* Known positive test for human immunodeficiency virus (HIV)
* Subjects with acute hepatitis B and C are not eligible. Subjects with chronic hepatitis B or C are eligible if their condition is stable and, in the opinion of the investigator and Amgen physician, if consulted, would not pose a risk to subject safety
* Toxicities from prior anti-tumor therapy not resolved to Common Terminology Criteria for Adverse Events (CTCAE) grade 0 or 1, or to levels dictated in the eligibility criteria. Grade 2 toxicities from prior anti-tumor therapy that are considered irreversible (defined as having been present and stable for \> 4 weeks), such as stable grade 2 peripheral neuropathy or ifosfamide-related proteinuria, may be allowed if they are not otherwise described in the exclusion criteria AND there is agreement to allow by the principal investigator
* Prior participation in an investigational study and/or procedure within 21 days of study day 1
* Anti-tumor therapy within 21 days of study day 1 including chemotherapy, antibody therapy, or other investigational agent. Enrollment of subjects that have received molecularly targeted agents less than 28 days prior to study day 1 will be permitted if more than 14 days and at least 5 drug half-lives have passed prior to receiving the first dose of HCQ
* Major surgery within 28 days of study day 1
* Any co-morbid medical disorder that may increase the risk of toxicity, in the opinion of the investigator
* All herbal supplements, vitamins, and nutritional supplements taken within the last 30 days prior to dosing on day 1 (and continued use, if appropriate), must be reviewed and approved by the principal investigator (PI)
* Patients with pre-existing maculopathy or retinopathy of the eye