Genetically Modified T Cells and Decitabine in Treating Patients With Recurrent or Refractory Ovarian, Primary Peritoneal, or Fallopian Tube Cancer

Inclusion Criteria:

* Patients with recurrent or refractory epithelial or non-epithelial ovarian, primary peritoneal or fallopian tube carcinoma who have received platinum containing chemotherapy and either has platinum refractory or resistant disease, or if plantinum sensitive disease, have received \>= 2 lines of chemotherapy. Subjects may have received PARP inhibitators , bevacizumab or immunotherapy. Non-epithelial tumors of the ovary include sarcomas, granulosa cell tumors and malignant germ cell tumors including chiriocarcinoma
* Have been informed of other treatment options
* Must be HLA- A\*02;01 positive; retesting is not required for patients who have previous documented HLA-A\*02;01 positivity
* Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Life expectancy of \> 4 months
* At least 4 weeks from prior chemotherapy, radiotherapy or immunotherapy, or prior investigational agents
* Must have measurable disease as defined by irRECIST
* Must have adequate venous access for apheresis; (pheresis catheter placement for cell collection is allowed)
* Women of childbearing potential in agreement to use acceptable birth control methods for the duration of the study and until persistence of the study drug is no longer detected in the peripheral blood; this may be a period of several years; methods for acceptable birth control include: condoms, diaphragm or cervical cap with spermicide, intrauterine device, and hormonal contraception; it is recommended that a combination of two methods be used
* Leukocytes \>= 3 x 10\^9/L
* Absolute neutrophil count \>= 1 x 10\^9/L
* Platelets \>= 100 x 10\^9/L
* Total bilirubin within normal institutional limits
* Aspartate Aminotransferases (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal
* Creatinine level = 2X\< upper limit of normal (ULN): if creatinine \> 2XULN, creatinine clearance must be \> 60ml/min
* Patient must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure

Exclusion Criteria:

* Patients receiving any other investigational agents
* Patients with active brain metastases should be excluded from this clinical trial; patients with prior history of brain metastasis who have undergone local therapy (i.e., metastasectomy and/or radiation) and show no evidence of local recurrence or progression over the past 6 months are eligible. Brain MRI as clinically indicated only
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to cyclophosphamide, decitabine or other agents used in the study
* Prior malignancy (except non melanoma skin cancer) within 3 years
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, or psychiatric illness/social situations that would limit compliance with study requirements
* Use of chronic corticosteroids, hydroxyurea, or immunomodulating agents (e.g., interleukin 2, interferon alpha or gamma, granulocyte colony stimulating factors, etc.) within 30 days prior to study entry

  * NOTE: recent or current use of inhaled steroids is not exclusionary; if subjects are prescribed a brief course of oral corticosteroids, the use should be limited to less than 7 days
* Active infection with human immunodeficiency virus (HIV), hepatitis B virus (HBV), hepatitis C virus (HCV), or cytomegalovirus (CMV) as defined below, due to the immunosuppressive effects of cyclophosphamide used and the unknown risks associated with viral replication

  * Positive serology for HIV
  * Active hepatitis B infection as determined by a positive test for hepatitis B surface antigen (Ag)
  * Active hepatitis C; patients will be screened for HCV antibody; if the HCV antibody is positive, a screening HCV ribonucleic acid (RNA) by any real time polymerase chain reaction (RT PCR) or branched deoxyribose nucleic acid (bDNA) assay must be performed at screening by a local laboratory with a Clinical Laboratory Improvement Act (CLIA) certification or its equivalent; eligibility will be determined based on a negative screening value; the test is not required if documentation of a negative result of a HCV RNA test performed within 60 days prior to screening is provided
  * Serology (CMV immunoglobulin G \[IgG\]) positive for active CMV
* Received any previous gene therapy using an integrating vector within 6 months
* Pregnancy or breast-feeding
* Lack of availability of a patient for immunological and clinical follow up assessment
* Evidence or history of significant cardiac disease (including evidence or history of significant cardiac disease (including myocardial infarction \[MI\] in the past 6 months, significant cardiac arrhythmia, stage III or IV congestive heart failure \[CHF\]); cardiac stress test will be done as clinically indicated; (the specific test to be chosen at the discretion of the principal investigator \[PI\])
* Patients with pulmonary function test abnormalities as evidenced by a forced expiratory volume in 1 second to forced vital capacity ratio measurement (FEV1/FVC) \< 70% of predicted for normality will be excluded