In Middle School I went to Wellington which, at least at the time, was the most prestigious Private School in Columbus.
The Grade School was an old abandoned Upper Arlington Elementary School that had been shuttered for decades. The Middle School was the original Perry Township school building that had been constructed decades before Upper Arlington even existed. The buildings were old. They were gross. They were falling apart. There was no AC. They smelled terrible. The teachers made a LOT less than their public school counterparts. About a fifth of the student body were poor inner city black kids that went there for free because they were good students and nice kids (I had a free ride as well).
It doesn't cost a lot to run a good school, if the students, the teachers, and the parents are invested in making it so.