High School on military post?

DODDS runs schools overseas. Unless there's a large distance, the Pols won't allow it because the service is provided out in town and the military is taking food out of babies mouths in doing so. The unions would go berserk as well as pulling Federal Impact Funds out of the school districts. Pretty much a non starter if there is a servicing school district around. DoD is under strict orders to utilize outside services.

Some large installations especially overseas, have them. In most cases in CONUS, there wouldn't be enough students to justify one on post (which would mean generally open only to the military community due to access issues) Many do have elementary and middle schools. Travis AFB has a district which primarily serves the Travis AFB base community (separate district from Fairfield (CA) Unified School District. It has their high and middle school and 3 elementary just off base with 2 elementary on-base. The district boundaries include parts of Fairfield and Vacaville nearer to the base. You need to have no military affiliation to attend those district off-base schools if you live in the district boundary. (Military families outside the district boundaries can attend that districts schools since they cater to the military community.)The 2 elementary schools on base service those families that live on base.

There are HS on post. At Randolph AFB in San Antonio I pass by it every morning.