Hydroxychloroquine, Cyclophosphamide, Dexamethasone, and Sirolimus in Treating Patients With Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * Histologically confirmed multiple myeloma * Documented relapse or persistent disease after at least 1 prior therapy containing both bortezomib and lenalidomide; or at least 2 prior therapies containing bortezomib in one and lenalidomide in the other, or if intolerant of bortezomib and/or lenalidomide; prior autologous and allogeneic bone marrow transplantation are allowed * Need for further treatment for myeloma, as determined by the patient's treating physician; this is defined as progression of clinical features (worsening anemia, renal function, bone disease, hypercalcemia, recurrent infections, and constitutional symptoms) OR biochemical progression (increasing M-spike in serum or urine, involved serum or urine free light chain over 2 consecutive time points greater than 4 weeks apart) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Ability to understand and the willingness to sign a written informed consent document * Birth control is required with full barrier contraceptives or complete abstinence for the duration of time receiving therapy and for 6 months after completing the last drug taken * The need for further treatment: this is defined as progression of clinical features (worsening anemia, renal function, bone disease, hypercalcemia, recurrent infections, and constitutional symptoms) OR biochemical progression (increasing M-spike in serum or urine, involved serum or urine free light chain over 2 consecutive time points greater than 4 weeks apart) Exclusion Criteria: * History of allergic reactions to compounds of similar chemical or biological composition to rapamycin or hydroxychloroquine * Patients may not take any of the following medications while on study, but will be considered eligible if medication is discontinued at least 72 hours (hrs) prior to first dose of Rapamycin: * Carbamazepine * Rifabutin * Rifampin * Rifapentine * St. John's wort * Clarithromycin * Cyclosporine * Diltiazem * Erythromycin * Itraconazole * Fluconazole * Ketoconazole * Telithromycin * Verapamil * Voriconazole * Posaconazole * Known macular degeneration or retinopathy (diabetic or otherwise), porphyria, or psoriasis (well-controlled psoriasis allowed provided under the care of a specialist who agrees to monitor the patient for exacerbations) * Absolute neutrophil count (ANC) =\< 1.0 x 10\^9/L * Platelets =\< 50 x 10\^9/L for any reason * Serum creatinine \>= 2.5 mg/dL * Total or direct bilirubin \>= 2.0 mg/dL * Transaminases 2 x the upper limit of normal * Fasting glucose \>= 200 mg/dL * Serum potassium \< 3.4 mmol/l * Serum phosphorus \< 2.4 mg/dl * Other conditions that would require therapy with hydroxychloroquine, including but not limited to, any of the following: * Systemic lupus * Rheumatoid arthritis * Porphyria cutanea tarda * Malaria treatment or prophylaxis * Evidence of other active malignancy, except: * Basal cell or squamous cell carcinoma of the skin * Treated carcinoma in situ * Uncontrolled intercurrent illness including, but not limited to, any of the following: * Uncontrolled ongoing infection * Human immunodeficiency virus (HIV) * Hepatitis B infection * Known glucose-6-phosphate dehydrogenase (G6PD) deficiency * Symptomatic congestive heart failure * Unstable angina pectoris * Uncontrolled cardiac arrhythmia * Psychiatric illness or social situations that would limit compliance with study requirements * Active graft-versus-host disease (GvHD) * Inability to understand or unwillingness to sign the informed consent document * Concurrent anti-myeloma therapy within: * 7 days of prior corticosteroids * 14 days of prior antimyeloma agents, including thalidomide or lenalidomide * 28 days of a different investigational regimen * 14 days of any radiation * Women of child-bearing who are unwilling or unable to use an acceptable method to avoid pregnancy for the entire study period and for up to 30 days after the last dose of study drug * Women who are pregnant or breastfeeding * History of G6PD deficiency * Known history of HIV infection