• PHASE 1: Patients must have a histologically confirmed metastatic or locally advanced unresectable solid tumor that has progressed on or after available standard of care therapy or for which no acceptable standard of care therapy exists, or in which the patient declines standard of care therapy (each patient that declines standard of care therapy will be documented in the case report form)

• PHASE 2: Patients must have a histologically confirmed metastatic or locally advanced unresectable cholangiocarcinoma/gallbladder carcinoma that has progressed on gemcitabine, cisplatin, and durvalumab/pembrolizumab.

• Age \>= 18 years

⁃ Because no dosing or adverse event data are currently available on the use of peposertib (M3814) in combination with avelumab in patients \< 18 years of age

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

• Patients with at least 1 index lesion to irradiate for whom palliative radiation treatment is indicated (including but not limited to pain and/or symptom control, prevention of disease -related complications, and preservation of organ function). Lung and liver lesions are preferred, though alternate lesions may be considered after discussion with trial principal investigator (PI). Up to 2 lesions may be considered for irradiation provided at least 1 lesion will receive the study treatment of total of 60 Gy and all prescribed irradiation will be completed within the radiation window

• Patients with at least 1 Response Evaluation Criteria in Solid Tumors (RECIST) measurable lesion (to be unirradiated) (defined as those accurately measured in at least one dimension, with the longest diameter to be recorded for non-nodal lesions and the shortest diameter for nodal lesions). Measurable is defined as at least 10 mm in longest diameter for solid tumors, at least 15 mm in shortest diameter for lymph nodes

• Patients must be willing to undergo fresh biopsies at baseline (as opposed to using archival tissue), in the event their baseline tissue was obtained \> 12 months prior to study consent and/or they are randomized to the gamma H2AX, pNBS1 and pKAP1 IFA with beta CATN segmentation assay

• Absolute neutrophil count (ANC) \>= 1,500/mcL

• Platelet count \>= 100,000/mcL

• Hemoglobin \>= 9.0 g/dL

• Serum creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated serum creatinine clearance (glomerular filtration rate \[GFR\] can be used in place of creatinine or creatinine clearance) \>= 60 mL/min for participants with creatinine levels \> 1.5 x institutional ULN

⁃ Calculate serum creatinine clearance using the standard Cockcroft-Gault formula

• Serum total bilirubin =\< 1.5 x ULN or direct bilirubin =\< ULN for participants with total bilirubin \> 1.5 x ULN

⁃ Patients with known Gilbert disease with serum bilirubin level =\< 3 x ULN are eligible

• Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x ULN or =\< 5.0 x ULN for patients with hepatobiliary tumors/liver metastases

• Albumin \>= 2.8 g/L

• International normalized ratio (INR) or prothrombin time (PT) or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN

⁃ This applies only to patients not receiving therapeutic anticoagulation; patients receiving therapeutic anticoagulation should be on a stable dose

• Participants must have the ability to swallow and retain oral medication and not have any clinically significant gastrointestinal abnormalities that might alter absorption

• Female patients of childbearing potential must have a negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. The effects of peposertib (M3814) and avelumab on the developing human fetus are unknown and there is the potential for teratogenic or abortifacient effects. For this reason, women and men 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 treatment, and for 6 months after completion of peposertib (M3814) and avelumab 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. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with peposertib (M3814) and avelumab, breastfeeding should be discontinued if the mother is treated with peposertib (M3814) and avelumab

• Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a close caregiver or legally authorized representative (LAR) and/or family member available will also be eligible