Sirolimus and Auranofin in Treating Patients With Advanced or Recurrent Non-Small Cell Lung Cancer or Small Cell Lung Cancer

Inclusion Criteria: * Histologic or cytologic confirmation of lung cancer (squamous, ras-mutated adenocarcinoma or small cell lung cancer) * Patients must have received at least one course of chemotherapy consisting of a platinum doublet and must have no acceptable standard treatment options * Prior radiation therapy is permitted as long as: * Recovered from the toxic effects of radiation treatment before study entry, except for alopecia * Absolute neutrophil count (ANC) \>= 1500 uL * Platelets (PLT) \>= 100,000 uL * Hemoglobin (Hgb) \>= 9 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) or direct bilirubin =\< ULN * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamic pyruvic transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 3 x ULN or SGOT (AST) and SGPT (ALT) =\< 5 x ULN is acceptable if liver has tumor involvement * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, 2 * Negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Ability to provide informed consent * Life expectancy \>= 12 weeks * Willing to return to Mayo Clinic enrolling institution for follow-up * Willing to provide tissue samples for correlative research purposes Exclusion Criteria: * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Symptomatic, untreated, or uncontrolled central nervous system (CNS) metastases or seizure disorder; NOTE: patients with treated CNS metastases without evidence of progression and without uncontrolled symptoms or need for steroids may enroll * Human immunodeficiency virus (HIV)-positive patients receiving combination anti-retroviral therapy are excluded * Unwilling or unable to, comply with the protocol * Any of the following prior therapies: * Radiation to \>= 25% of bone marrow * Major surgery (i.e., laparotomy), open biopsy, or significant traumatic injury =\< 4 weeks prior to registration; minor surgery =\< 2 weeks prior to registration; insertion of a vascular access device is not considered major or minor surgery in this regard * Any of the following concurrent severe and/or uncontrolled medical conditions: * Hypertension, labile hypertension, or history of poor compliance with antihypertensive medication * Angina pectoris * History of congestive heart failure =\< 3 months, unless ejection fraction \> 40% * Myocardial infarction =\< 6 months prior to registration * Cardiac arrhythmia * Poorly controlled diabetes * Interstitial pneumonia or extensive and symptomatic interstitial fibrosis of the lung * Active or recent history of hemoptysis; if hemoptysis has resolved with measures such as palliative radiation therapy (e.g. 3000 cGy over 10 fractions), arteriographic embolization or endobronchial interventions (e.g. photodynamic therapy, brachytherapy), etc. for \> 14 days, patients may be considered for participation in this study * \>= Grade 2 hypertriglyceridemia * \>= Grade 2 hypercholesterolemia * Any illness that in the opinion of the investigator would compromise the ability of the patient to participate safely in the clinical trial * Use of St. John's Wort because of its effects on hepatic drug metabolism * Other active malignancy: EXCEPTIONS: Non-melanoma skin cancer, localized prostate cancer, or carcinoma-in-situ of the cervix; NOTE: If there is a history or prior malignancy, patient must not be receiving other cytotoxic or molecularly targeted therapeutics treatment for their cancer; patients receiving certain hormonal manipulations as part of their treatment may be allowed to continue at the discretion of the Principal Investigator (PI) (e.g. luteinizing hormone-releasing hormone \[LHRH\] analogs for prostate cancer) * Unable to discontinue use of potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inhibitors/inducers