Testing the Combination of Two Anti-cancer Drugs, Peposertib (M3814) and Tuvusertib (M1774) for Advanced Solid Tumors

Inclusion Criteria: * Patients must have histologically confirmed solid malignancy that is metastatic or unresectable and for which standard curative or palliative measures do not exist or are no longer effective, or if the treating investigator deems the study appropriate. * For the dose escalation and dose expansion phases, patients must have genomic (tumor next-generation sequencing \[NGS\], circulating tumor deoxyribonucleic acid \[ctDNA\], fluorescence in situ hybridization \[FISH\], etc.) or immunohistochemical evidence (e.g., loss of expression) of inactivating ATM mutations, MYC amplification, mutation of FBXW7, CCNE1 amplification, SWI/SNF member mutation (ARID1A, PBRM1, SMARCA4, ARID2, ARID1b, SMARCB1, SMARCA2, SS18) and mutation or loss of expression of ATRX/DAXX. Mutations may be germline or somatic. All mutations/alterations must be approved by the overall principal investigator (PI). Other SWI/SNF mutations may be considered after discussion with the overall PI. * Progression on at least one prior standard therapy (if no standard therapy exists, the patient may be allowed if the treating investigator deems appropriate). * Age \>= 18 years. Because no dosing or adverse event data are currently available on the use of peposertib (M3814) in combination with tuvusertib (M1774) in patients \< 18 years of age, children are excluded from this study. * Life expectancy \> 3 months. * Eastern cooperative oncology group (ECOG) performance status =\< 2 (Karnofsky \>= 60%). * Measurable disease by response evaluation criteria in solid tumors (RECIST) 1.1 (RECIST 1.1 non-measurable disease permitted for the dose escalation portion). * Hemoglobin \>= 9 g/dL. * Absolute neutrophil count \>= 1,500/mcL. * Platelets \>= 100,000/mcL. * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN). * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]/alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase \[SGPT\]) =\< 3 × institutional ULN or =\< 5.0X the ULN if liver metastases are present. * Glomerular filtration rate (GFR) \>= 60 mL/min/1.73m\^2. * Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial. Anti-retroviral therapy agents must be considered for potential drug-drug interactions per exclusion. * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured or have evidence of clearance of HCV (i.e., undetectable HCV viral load). For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better. * Able to swallow whole capsules or tablets. * Willing to undergo paired biopsies (expansion arm); if a biopsy is not feasible or safe, the patient may be allowed to participate after discussion with the overall PI. * Female patients of childbearing potential must have a negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. * Female patients of childbearing potential must be willing to use an adequate method of contraception for the course of the study through 6 months after the last dose of study medication. * Male patients of reproductive potential must agree to avoid impregnating a partner while receiving study drug and for 3 months after the last dose of study drug by complying with adequate methods of contraception. * Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the patient. * Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity who have a legally-authorized representative (LAR) and/or family member available will also be eligible. Exclusion Criteria: * Patients who have received immunotherapy within 21 days of Cycle 1 Day 1. * Patients who have received therapeutic radiation therapy within 21 days, or palliative radiation therapy within 7 days, of Cycle 1 Day 1. * Patients who have undergone major surgery within 21 days of Cycle 1 Day 1. * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> Grade 1) with the exception of alopecia, controlled endocrine toxicity (e.g., hypothyroidism), and cutaneous toxicity which will be permitted at Grade 2. * Patients who are receiving any other investigational agents. * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate central nervous system (CNS) specific treatment is not required and is unlikely to be required during the first cycle of therapy. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to peposertib (M3814) and tuvusertib (M1774). * Patients who cannot discontinue concomitant medications or herbal supplements that are strong inhibitors or strong inducers of cytochrome P450 (CYP) isoenzymes, CYP1A2, CYP3A4/5, CYP2C19, and CYP2C9. Concomitant use of substrates hMATE1, hMATE2, and CYP3A4/5 substrates with a narrow therapeutic index are also excluded. Opioids may interact with these enzymes; use of opioids while on study is allowed but should be closely monitored. Concomitant administration of sensitive substrates of P-gp, BCRP, OCT1, OATP1B1, and OATP1B3 should be avoided (if the use is unavoidable, carefully monitor patients for signs of increased toxicity). Patients may confer with the study doctor to determine if alternative medications can be used. The following categories of medications and herbal supplements must be discontinued for at least the specified period of time before the patient can be treated: * Strong inducers of CYP1A2, CYP3A4/5, CYP2C19, and CYP2C9: \>= 3 weeks prior to study treatment. * Strong inhibitors of CYP1A2, CYP3A4/5, CYP2C19, and CYP2C9: \>= 1 week prior to study treatment. * Substrates of hMATE1, hMATE2, CYP3A4/5, P-gp, BCRP, OCT1, OATP1B1, and OATP1B3 with a narrow therapeutic index: \>= 1 day prior to study treatment. * Patients who cannot discontinue proton-pump inhibitors (PPIs). H-2-receptor antagonist should be held during the 2 weeks of concurrent dosing with peposertib (M3814). There is no H-2-receptor antagonist restriction during the off weeks without peposertib (M3814) dosing. H-2-receptor antagonists should not be taken within 12 hours before or 2 hours after tuvusertib (M1774). Antacids should not be taken within 2 hours before or 2 hours after tuvusertib (M1774). * Patients who received hematopoietic growth factor (e.g., granulocyte colony-stimulating factor, erythropoietin) within 14 days prior to the first dose of study intervention. * Patients with uncontrolled intercurrent illness including, but not limited to: ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or cardiac arrhythmia. * QTcF (using the Fridericia correction calculation) of \>= 470 msec * Pregnant women and women who are breastfeeding are excluded from this study because the effects of the study drugs on the developing fetus are unknown. * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen as assessed by the treating investigator may be included with the approval of the sponsor-investigator.