Fighting Homelessness Through Services and Housing Act This bill requires the Health Resources and Services Administration to award grants to local and tribal government entities for the planning and implementation of programs to address homelessness. The program includes five-year implementation grants to assist such entities in carrying out activities and paying capital building costs associated with the provision of housing and services to homeless individuals and families. Eligible entities must demonstrate the capacity to address areas such as mental health, substance use disorder and recovery services, chronic health conditions, and educational and job training.