In our starting years, we built at the school. Generally we had a small-ish storage room for tools and parts and we used a classroom after school for building, but everything had to be put back in the store room.
A couple years later, we got a larger room to work in Motorola, about the size of a small classroom, but the adjacent presentation hall was almost never occupied and we had a computer room. So we had A LOT of space at the motorola facility.
Then after realizing that some people could not make it to the facility, we moved back to school. We have the same, small storage space to work, but we have a small machine shop and two rooms after school and on weekends.
EDIT: I totally didn't answer your question.
. Sponsors can be a big help with space and machine shop work, but if not, then the school is a good bet. I would say that if you need more room, ask your sponsor to see what's available after school. Don't get caught on the fact that you'd have to clean up after each build. We did that almost every day for 6 weeks, so you'll have to compromise.