Genetically Engineered PBMC and PBSC Expressing NY-ESO-1 TCR After a Myeloablative Conditioning Regimen to Treat Patients With Advanced Cancer

Inclusion Criteria: * Stage IV or locally advanced unresectable cancers for which no alternative therapies with proven survival advantage are available * NY-ESO-1 positive malignancy by immunohistochemistry (IHC) utilizing commercially available NY-ESO-1 antibodies * HLA-A\*0201 (HLA-A2.1) positivity by molecular subtyping * Age greater than or equal to 16 years old; if patients 16-17 years old are enrolled in the trial, they will only be enrolled after 3 patients \>= 18 years old have been treated, and the treatment has been shown to be safe * A minimum of one measurable lesion defined as: * Meeting the criteria for measurable disease according to Response Evaluation Criteria in Solid Tumors (RECIST) * Skin lesion(s) selected as non-completely biopsied target lesion(s) that can be accurately measured and recorded by color photography with a ruler to document the size of the target lesion(s) * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0 or 1 * Adequate bone marrow and major organ function to undergo a PBSC transplant determined within 30-60 days prior to enrollment using standard phase 1 criteria for organ function defined as: * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9 cells/L * Platelets \>= 100 x 10\^9/L * Hemoglobin \>= 9 g/dL * Aspartate and alanine aminotransferases (AST, ALT) =\< 2.5 x ULN (upper limit of normal) (=\< 5 x ULN, if documented liver metastases are present) * Total bilirubin =\< 2 x ULN (except patients with documented Gilbert?s syndrome) * Creatinine \< 2 mg/dl (or a glomerular filtration rate \> 60) * Must be willing and able to accept at least three leukapheresis procedures * Must be willing and able to undergo three research PET scans * Must be willing and able to provide written informed consent Exclusion Criteria: * Inability to purify \>= 2.5 x 10\^6 CD34-enriched cells/kg of patient weight from the pooled G-CSF mobilized leukapheresis products * Previously known hypersensitivity to any of the agents used in this study; known sensitivity to busulfan or fludarabine * Received systemic treatment for cancer, including immunotherapy, within 28 days prior to initiation of conditioning chemotherapy administration within this protocol * Potential requirement for systemic corticosteroids or concurrent immunosuppressive drugs based on prior history or received systemic steroids within the last 2 weeks prior to enrollment (inhaled or topical steroids at standard doses are allowed) * Human immunodeficiency virus (HIV) seropositivity or other congenital or acquired immune deficiency state, which would increase the risk of opportunistic infections and other complications during chemotherapy-induced lymphodepletion; if there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Hepatitis B or C seropositivity with evidence of ongoing liver damage, which would increase the likelihood of hepatic toxicities from the chemotherapy conditioning regimen and supportive treatments; if there is a positive result in the infectious disease testing that was not previously known, the patient will be referred to their primary physician and/or infectious disease specialist * Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol * Known clinically active brain metastases; prior evidence of brain metastasis successfully treated with surgery or radiation therapy will not be exclusion for participation as long as they are deemed under control at the time of study enrollment and there are no neurological signs of potential brain metastases * Pregnancy or breast-feeding; female patients must be surgically sterile or be postmenopausal for two years, or must agree to use effective contraception during the period of treatment and for 6 months afterwards; all female patients with reproductive potential must have a negative pregnancy test (serum/urine) within 14 days from starting the conditioning chemotherapy; the definition of effective contraception will be based on the judgment of the study investigators * Since IL-2 is administered following cell infusion: * Patients will be excluded if they have a history of clinically significant electrocardiogram (ECG) abnormalities, symptoms of cardiac ischemia with evidence of ischemia on a cardiac stress test (stress thallium, stress multigated acquisition \[MUGA\], dobutamine echocardiogram or other stress test) * Similarly, patients with a baseline left ventricular ejection fraction (LVEF) \< 45% will be excluded. * Patients with ECG results of any conduction delays (PR interval \> 200 ms, corrected QT \[QTC\] \> 480 ms), sinus bradycardia (resting heart rate \< 50 beats per minute), sinus tachycardia (heart rate \>120 beats per minute) will be evaluated by a cardiologist prior to starting the trial; patients with any arrhythmias, including atrial fibrillation/atrial flutter, excessive ectopy (defined as \> 20 premature ventricular contractions \[PVCs\] per minute), ventricular tachycardia or 3rd degree heart block will be excluded from the study unless cleared by a cardiologist * Patients with pulmonary function test abnormalities as evidenced by a (forced expiratory volume 1 \[FEV1\]/forced vital capacity \[FVC \] \< 70% of predicted for normality will be excluded * Bone marrow involvement based on PET/CT scan at screening * Active or recent herpes simplex virus (HSV) infection or cytomegalovirus (CMV) based on symptoms with positive swab culture and/or positive IgM (immunoglobulin M) screening * Liver metastases with no other metastatic sites