Enzalutamide and PDS01ADC in PET Positive Recurrent Prostate Cancer (pprPC) Without Testosterone Lowering Therapy

* INCLUSION CRITERIA: * Participant must provide documentation of histologic or cytological confirmation of prostate cancer or tumor sample for diagnosis confirmation. Note: in the absence of pathology or documentation, participant must have a rising PSA, PSMA+ disease, and his history consistent with prostate cancer as documented by the investigator. * History of primary treatment for prostate cancer (either surgery or radiation). * Prostate-specific antigen (PSA) doubling time within less than 12 months. * Testosterone \>100 ng/dL. * Age \>=18 years. * Evidence of prostate cancer on PSMA PET/CT scan. * Eastern Cooperative Oncology Group (ECOG) performance status \<2. * Men must agree to use an effective method of contraception (barrier or surgical sterilization) after study entry and for 3 months after completion of enzalutamide or PDS01ADC therapy whatever comes later. * Participants must have adequate organ and marrow function as defined below: * Absolute neutrophil count (ANC) \>=1,500/microliter, without granulocyte colony-stimulating factor (G-CSF) support * Platelets \>=100,000/microliter * Aspartate aminotransferase (AST) /Alanine aminotransferase (ALT) \<=2.5 x institutional upper limit of normal (ULN) * Hemoglobin (Hgb) \>= 10 g/dL (packed red blood cell (pRBC) transfusions are not allowed to achieve acceptable Hgb) * Total bilirubin \<= 1.5 x ULN, OR \<= 3.0 ULN in participants with Gilbert s syndrome * Serum albumin \>= 2.8 g/dL * Creatinine \< 1.5 X institution ULN OR --Measured or calculated creatinine clearance (CrCl) (estimated glomerular filtration rate (eGFR) may also be used in place of CrCl) \> 45 mL/min/1.73 m\^2 for participant with creatinine levels \> 1.5 x institutional ULN * Hepatitis B virus (HBV)-infected participants can be enrolled if HBV DNA is undetectable at screening. Hepatitis C virus (HCV)-infected participants can be enrolled if the HCV RNA level is undetectable at screening. Human immunodeficiency virus (HIV)-positive participants can be enrolled if HIV DNA is undetectable. * Participants must be able to swallow tablets/capsules. * Participants must be able to understand and willing to sign a written informed consent document. EXCLUSION CRITERIA: * Evidence of soft tissue disease on CT scan (or magnetic resonance imaging (MRI) if assessment cannot be done by CT scan) per RECIST 1.1 criteria (lymph nodes up to 2.0 cm in the shortest dimension are allowed). * Evidence of bone lesions on Tc99 bone scan. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to study drugs or imaging agents used in the study. * Any medical condition that requires chronic systemic steroid therapy, or any other form of immunosuppressive medication (inhaled and topical steroids are permitted). * History of seizures within the last 10 years. * Therapy with strong inhibitors or inducers of CYP2C8 or CYP3A4 (https://druginteractions.medicine.iu.edu/MainTable.aspx) within 5 half-lives prior to the study treatment initiation. * Participants with prior malignancy active within 3 years prior to study treatment initiation except for locally curable cancers that have been apparently cured such as basal or squamous cell skin cancer, superficial bladder cancer, or carcinoma in situ of the breast. * Uncontrolled intercurrent illness that would limit compliance with study requirements.