Berzosertib and Irinotecan in Treating Patients With Progressive, Metastatic, or Unresectable TP53 Mutant Gastric or Gastroesophageal Junction Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed progressive metastatic or unresectable gastric or GEJ adenocarcinoma. * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) by chest x-ray or as \>= 10 mm (\>= 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam. * Patients must have progressed after or been intolerant of at least two lines of systemic therapy. Patients with HER2 positive gastric and GEJ adenocarcinoma must have progressed on trastuzumab plus chemotherapy in the first line setting. Patients with microsatellite unstable (MSI-H) tumors must have received prior immunotherapy with pembrolizumab. * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of M6620 in combination with irinotecan in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * Both men and women of all races and ethnic groups are eligible for this trial. * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 60%). * Leukocytes \>= 3,000/mcL. * Absolute neutrophil count \>= 1,500/mcL. * Platelets \>= 100,000/mcL. * Hemoglobin \>= 9 g/dL. * Total bilirubin within normal institutional limits. * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal (ULN); if liver involvement =\< 5 x ULN. * Creatinine clearance \>= 45 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal. * Patients must have a TP53 mutation (only those known hot-spot mutations that fall within exon 2 or exons 4-11 will be accepted) determined from available archived tumor tissue that has been subjected to next generation sequencing (NGS) through FoundationOne/FoundationOneCDx or a similar assay performed in a Clinical Laboratory Improvement Amendments (CLIA)-certified laboratory. Investigators from other sites, who have potential patients who meet study eligibility, will send copies of NGS reports from these patients via Medidata Rave case reports to the responsible study coordinator. Our research team will review each report to ensure each patient possesses the mutations of interest. Similar review will happen for each patient we enroll on the study at our institution. Case reports from all screened patients will be centrally available on the Rave study database. * Nine patients must be willing to undergo endoscopic or CT guided tumor biopsies for mandatory correlative studies. If the biopsy is deemed not safe by the treating physician, the patient may still enroll given that the other eligibility criteria are met. * The effects of M6620 on the developing human fetus are unknown. For this reason and because deoxyribonucleic acid (DNA)-damage response (DDR) inhibitors, as well as irinotecan, are known to be teratogenic, women of child-bearing potential and men able to father children who have female partners of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 6 months after trial participant's final dose of M6620 or irinotecan (whichever agent is completed last). Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Patients with early stage untreated or resectable gastric adenocarcinoma. * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. * Patients who have previously received irinotecan. * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1, except alopecia) that was administered more than four weeks prior to starting study therapy. * Patients who are receiving any other investigational agents. * Patients with untreated or symptomatic brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M6620 or irinotecan. * M6620 is primarily metabolized by CYP3A4, and irinotecan and its active metabolite, SN-38, are metabolized by CYP3A4 and UGT1A1, respectively; therefore, concomitant administration with strong inhibitors of CYP3A4 (e.g., ketoconazole, itraconazole, clarithromycin, ritonavir, indinavir, nelfinavir and saquinavir) or inducers of CYP3A4 (e.g. rifampin, phenytoin, carbamazepine, phenobarbital, St. John's Wort) should be avoided. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference for a list of drugs to avoid or minimize use of. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. * 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. * Pregnant women are excluded from this study because M6620 as a DNA-damage response (DDR) inhibitor may have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with M6620, breastfeeding should be discontinued if the mother is treated with M6620. These potential risks also apply to irinotecan. * Human immunodeficiency virus (HIV)-positive patients are excluded unless they have an undetectable viral load and are able to use anti-viral agents that do not interact with CYP3A4 (or regimens with agents that are not major inhibitors of cytochrome P450 enzymes). * History of other malignancy within 36 months prior to enrollment. Patients with local cancers of any type, provided no recurrence over this timeframe, are eligible.