• Patients must have histologically or cytologically confirmed diagnosis of metastatic advanced cancer.

• In dose escalation, any solid tumor patients with either O6-methylguanine DNA methyltransferase (MGMT) promoter hypermethylation positivity on testing / pre-screening of archival tissue OR an extracranial solid tumor where TMZ is considered a standard of care per National Comprehensive Cancer Network (NCCN) guidelines (neuroendocrine tumor, small cell lung cancer, melanoma or soft tissue sarcoma). The tumor lesion must be safely accessible to a mandatory biopsy. Patients with MGMT promoter hypermethylated colorectal cancer must be mismatch repair proficient / microsatellite stable.

• In phase 2, only patients with mismatch repair proficient / microsatellite stable colorectal cancer that have MGMT promoter hypermethylation positivity on pre-screening of archival tissue will be eligible.

• In dose escalation, patients must have progressed after treatment with all available therapies including immunotherapies for metastatic disease that are known to confer clinical benefit, or are intolerant to treatment, or refuse standard treatment. Patients may not have previously received temozolomide or an ataxia telangiectasia and rad3-related (ATR) inhibitor.

• For patients with mismatch repair proficient / microsatellite stable colorectal cancer in the phase 2 portion, patients must have received prior therapy with 1 or more systemic therapies in the metastatic setting that includes 5-fluorouracil, irinotecan, and oxaliplatin. Patients with microsatellite stable colorectal cancer (mCRC) need to have had exposure, unless contraindicated, to all 3 of oxaliplatin, irinotecan, and fluoropyrimidine (FP).

⁃ The use of 5-fluorouracil and oxaliplatin in the adjuvant setting is acceptable, provided the development of metastatic disease was less than 6 months after the completion of adjuvant therapy.

⁃ Patients with a prior hypersensitivity reaction to oxaliplatin in the adjuvant setting do not require retreatment in the metastatic setting.

• Age \>=18 years. Because no dosing or adverse event data are currently available on the use of M1774 in combination with temozolomide in patients \< 18 years of age, children are excluded from this study.

• Measurable disease on CT and/or MRI per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 criteria.

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (or Karnofsky \>= 60%).

• Hemoglobin \>=10 g/dL (No blood transfusions are allowed within 14 days of cycle 1 day 1 \[C1D1\]).

• White blood cells (WBC) \> 3 x 10\^9/L.

• 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 x institutional ULN except for when liver metastases are present, in which case they must be =\< 5 x institutional ULN.

• Glomerular filtration rate (GFR) \>= 60 mL/min/1.73 m\^2.

• Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial.

• 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. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load.

• Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression for 4 weeks.

• 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.

• The effects of M1774 on the developing human fetus are unknown. For this reason and because ATR inhibitors agents as well as other therapeutic agents used in this trial are known to be teratogenic, women 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 6 months after completion of M1774 administration. 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. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 3 months after completion of M1774 administration.

• Ability to understand and the willingness to sign a written informed consent document.