this was a private school or a public ?

also it says it was a high school on the headmaster's computer

he academy is probably named after Roosevelt High School, which is a real life school outside of Washington DC in Maryland. It is named after Eleanor Roosevelt rather than the two presidents Theodore and Franklin. Dean Dewey, the Academy's protectron, refers to the school as Roosevelt High School when activated at the Headmaster's terminal. It's doesn't say if it was a private school or a public school and I don't see would matter.