Elimusertib for the Treatment of Relapsed or Refractory Solid Tumors

Inclusion Criteria: * Part A: Patients between \>= 12 months and \< 18 years of age * Part B: * Patients between \>= 12 months and =\< 30 years of age for the phase 2 expansion cohorts for both EWS and PAX3-FOXO1 ARMS. * Patients between \>= 12 months and =\< 21 years of age for the phase 2 DDR expansion cohort * The Phase 2 cohorts will initially open concurrently with the Phase 1 portion but will only enroll patients at least 18 years of age. Patients \< 18 years of age will be included in the Phase 2 cohorts only after the RP2D/MTD has been estimated in the Phase 1 portion * All patients for both Parts A and B must have a minimum body surface area (BSA) \>= 0.74 m\^2 * All patients for both Parts A and B must have the ability to swallow BAY 1895344 (elimusertib) tablets intact * Patients with recurrent or refractory solid tumors. Patients must have had histologic verification of malignancy at original diagnosis or relapse * Part A: Any (non-CNS primary) solid tumor diagnosis including lymphoma which meets one of the following criteria: * Any Ewing Sarcoma (histological confirmation alone is adequate) or any EWS-fusion positive solid tumor (i.e. including related Ewing's family of tumors with EWS fusions such as EWS-WT1, EWS-ATF1, etc.) * Alveolar rhabdomyosarcoma (ARMS) with the PAX3-FOXO1 fusion. This does not include PAX7-FOXO1 or other variant fusion ARMS. Please note that a FISH showing FOXO1 breakapart is NOT sufficient for eligibility onto this cohort…