PSCA-CAR T Cells in Treating Patients With PSCA+ Metastatic Castration Resistant Prostate Cancer

Inclusion Criteria: * All participants must have the ability to understand and the willingness to sign a written informed consent * Eastern Cooperative Oncology Group (ECOG) performance status 0 - 2 or KPS ≥70%. * Documented castration resistant prostate cancer (mCRPC) (Note: castration will be defined by a testosterone \< 50 ng/dL achieved by orchiectomy or luteinizing hormone-releasing hormone \[LHRH\] agonist/antagonist therapy) * Documented PSCA+ tumor expression as evaluated by City of Hope (COH) Pathology Care * Progression of disease manifest by one of the following means during treatment with at least one advanced androgen targeted therapy (e.g., abiraterone or enzalutamide) * Rising PSA documented on 2 occasions at least 7 days apart, with absolute increase \> 2 ng/dL despite testosterone \< 50 OR * Radiographic evidence of new metastatic foci on computed tomography (CT) or bone scan, or soft tissue progression by Response Evaluation Criteria in Solid Tumors (RECIST) * Prior chemotherapy with cabazitaxel and/or docetaxel is allowed but not required. If there has been prior chemotherapy, at least 2 weeks must have elapsed prior to leukapheresis * Prior radiotherapy is allowed provided it was not administered to the only evaluable site of disease and was \> 14 days prior to leukapheresis * No known contraindications to leukapheresis, steroids or tocilizumab * Total serum bilirubin =\< 2.0 mg/dL (to be performed within 42 days of signing the main study consent) * Patients with Gilbert syndrome may be included if their total bilirubin is =\< 3.0 x upper limit of normal (ULN) and direct bilirubin =\< 1.5 x ULN * Aspartate aminotransferase (AST) \< 5 x ULN (to be performed within 42 days of signing the main study consent) * Alanine aminotransferase (ALT) \< 5 x ULN (to be performed within 42 days of signing the main study consent) * Creatinine clearance of \>= 50 mL/min per the Cockcroft-Gault formula (to be performed within 42 days of signing the main study consent) * Cardiac function (12 lead-electrocardiography \[ECG\]) without acute abnormalities requiring investigation or intervention (to be performed within 42 days of signing the main study consent) * Left ventricular ejection fraction \> 40% (to be performed within 42 days of signing the main study consent) * Participants of reproductive potential must agree to use acceptable birth control methods throughout study therapy and for 3 months after final dose of study treatment Exclusion Criteria: * Participants with clinically significant arrhythmia or arrhythmias not stable on medical management within two weeks of signing the main consent * Participants with a known history or prior diagnosis of optic neuritis or other immunologic or inflammatory disease affecting the central nervous system, including seizure disorder * History of allergic reactions attributed to compounds of similar chemical or biologic composition or other agents used in this study * Known bleeding disorders (e.g., von Willebrand's disease) or hemophilia * History of stroke or intracranial hemorrhage within 6 months prior to signing the main consent * History of other malignancies, except for malignancy surgically resected (or treated with other modalities) with curative intent, basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin; non-muscle invasive bladder cancer; malignancy treated with curative intent with no known active disease present for \>= 3 years * Uncontrolled active infection * Active hepatitis B or hepatitis C infection * Human immunodeficiency virus (HIV) infection * Any other condition that would, in the investigator's judgment, contraindicate the subject's participation in the clinical study due to safety concerns with clinical study procedures * Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)