Administering Peripheral Blood Lymphocytes Transduced With a Murine T-Cell Receptor Recognizing the G12V Variant of Mutated RAS in HLA-A*11:01 Patients

* INCLUSION CRITERIA: * Measurable (per RECIST V1.1 criteria, metastatic, or unresectable malignancy expressing G12V mutated KRAS as assessed by one of the following methods: RT-PCR on tumor tissue, tumor DNA sequencing, or any other CLIA-certified laboratory test on resected tissue. Patients shown to have tumors expressing G12V mutated NRAS and HRAS will also be eligible as these oncogenes share complete amino acid homology with G12V mutated KRAS for their first 80 N-terminal amino acids, completely encompassing the target epitope. * Patients must be HLA-A\*11:01 positive as confirmed by the NIH Department of Transfusion Medicine. * Confirmation of the diagnosis of cancer by the NCI Laboratory of Pathology. * Patients must have: * previously received standard systemic therapy for their advanced cancer and have been either non-responders or have recurred, specifically: * Patients with metastatic colorectal cancer must have had at least two systemic chemotherapy regimens that include 5FU, leucovorin, bevacizumab, oxaliplatin, and irinotecan (or similar agents), or have contraindications to receiving those medications. * Patients with pancreatic cancer must have received gemcitabine, 5FU, and oxaliplatin (or similar agents), or have contraindications to receiving those medications. * Patients with non-small cell lung cancer (NSCLC) must have had appropriate targeted therapy as indicated by abnormalities in ALK, EGFR, or expression of PDL- 1. Other patients must have had platinum-based chemotherapy. * Patients with ovarian cancer or prostate cancer must have had approved first-line chemotherapy. OR * declined standard treatment * Patients with 3 or fewer brain metastases that are less than 1 cm in diameter and asymptomatic are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month after treatment for the patient to be eligible. Patients with surgically resected brain metastases are eligible. * Age greater than or equal to 18 years and less than or equal to 72 years. * Clinical performance status of ECOG 0 or 1 * Patients must be willing to practice birth control from the time of enrollment on this study and 12 months after the last dose of combined chemotherapy for women and for 4 months after treatment for men. * Women of child-bearing potential must be willing to undergo pregnancy testing prior to the start of treatment because of the potentially dangerous effects of the treatment on the fetus. NOTE: Certain malignancies may secrete hormones that produce false positive pregnancy tests. Serial blood testing (e.g. HCG measurements) and/ or ultrasound may be performed for clarification. * Serology * Seronegative for HIV antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who are HIV seropositive may have decreased immune-competence and thus may be less responsive to the experimental treatment and more susceptible to its toxicities.) * Seronegative for hepatitis B antigen, and seronegative for hepatitis C antibody. If hepatitis C antibody test is positive, then patient must be tested for the presence of antigen by RT-PCR and be HCV RNA negative. * Hematology * ANC greater than 1000/mm\^3 without the support of filgrastim * WBC greater than or equal to 2500/mm\^3 * Platelet count greater than or equal to 80,000/mm\^3 * Hemoglobin \> 8.0 g/dL. Subjects may be transfused to reach this cut-off. * Chemistry * Serum ALT/AST less than or equal to 5.0 times ULN * Total bilirubin less than or equal to 2.0 mg/dL, except in patients with Gilbert s Syndrome, who must have a total bilirubin less than 3.0 mg/dL. * Patients must have either an eGFR \> 60 mL/m (based on serum creatinine and lab nomogram) or a formal 6-24h CrCl \> 60 mL/m. * Patients must have completed any prior systemic therapy at the time of enrollment. Note: Patients may have undergone minor surgical procedures or limited field radiotherapy within the four weeks prior to enrollment, as long as related major organ toxicities have recovered to less than or equal to grade 1. * Ability of subject to understand and the willingness to sign a written informed consent document. * Willing to sign a durable power of attorney. * Subjects must be co-enrolled on protocol 03C0277. EXCLUSION CRITERIA: * Large volume pulmonary irradiation. * Women of child-bearing potential who are pregnant or breastfeeding because of the potentially dangerous effects of the treatment on the fetus or infant. * Concurrent systemic steroid therapy. * Active systemic infections requiring anti-infective treatment, coagulation disorders, or any other active or uncompensated major medical illnesses. * Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). * Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who have decreased immune-competence may be less responsive to the experimental treatment and more susceptible to its toxicities.) * History of severe immediate hypersensitivity reaction to cyclophosphamide, fludarabine, or aldesleukin. * History of coronary revascularization or ischemic symptoms * For select patients with a clinical history prompting cardiac evaluation: last known LVEF less than or equal to 45%. * For select patients with a clinical history prompting pulmonary evaluation: known FEV1 less than or equal to 50% or DLCO less than 60%. * Patients who are receiving any other investigational agents.