Inclusion Criteria:
* Histologically confirmed adenocarcinoma of the prostate
* Recurrent prostate cancer following primary therapy as defined by:
* Post-radical prostatectomy: Any PSA \>= 0.4 ng/ml
* Post-primary radiotherapy: PSA \>= 2 ng/ml above a post-radiotherapy nadir
* Post-primary androgen-deprivation therapy: A confirmed rise of PSA \>= 2 ng/ml above a post-therapy nadir
* For patients with non-castrate levels of circulating androgen levels (testosterone \>= 50 g/dl)
* PSA levels should be increasing on at least two occasions \>= 1 week apart
* Patients should not be considered candidates for radiation therapy
* For patients with castrate levels of circulating androgen levels (testosterone \< 50 ng/dl):
* PSA levels must be \>= 0.4 ng/ml (if history of radical prostatectomy) or \>= 2 ng/ml (if history of non-surgical primary treatment) and found to be increasing on at least two occasions \>= 1 week apart
* At least 4 weeks must have elapsed since any changes to hormonal therapy, including at least 4 weeks since flutamide and at least 6 weeks since bicalutamide, nilutamide, or enzalutamide
* No evidence of metastatic cancer on imaging including a bone scan and computed tomography (CT) scan of chest/abdomen/pelvis
* Able to understand and adhere to dietary and medication restrictions as recommended for the safe use of phenelzine
* Men with child bearing potential are required to use an effective means of contraception
* Leukocytes \>= 3,000/mcL
* Absolute neutrophil count \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Total bilirubin =\< 1.5 x upper limit of normal (ULN) except in cases of benign isolated hyperbilirubinemia such as Gilbert's syndrome.
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SPGT\]) =\< 2.5 x ULN
* Creatinine =\< 1.5 x ULN
Exclusion Criteria:
* Uncontrolled hypertension despite appropriate medical therapy (blood pressure \[BP\] greater than 160 mmHg systolic and 90 mmHg diastolic at 2 separate measurements no more than 60 minutes apart during the screening visit); Note: patients may be rescreened after adjustment of antihypertensive medications
* Known prior history of mania or major psychiatric illness (schizophrenia, bipolar disorder, severe major depression requiring hospitalization, etc.)
* Concurrent use of medications contra-indicated due to potential interactions with phenelzine
* Inability to comply with dietary restrictions for foods, supplements, and medications with potential for adverse interactions with phenelzine or to otherwise cooperate fully with the investigator and study personnel
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to phenelzine or other monoamine oxidase inhibitors
* Patients may not be receiving any other investigational agents
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
