Sort of. I host a Matrix Synapse server for about 15 friends. Some of them use rooms on the big Matrix.org server as well, some bridge to Discord and Facebook, and some chat with other smaller groups. By default, Synapse won't sync rooms you haven't joined. It also won't copy files - it'll fetch those I believe (although maybe it'll keep a few days worth of files around?).
After about 5 years of hosting it, we're sitting at 90GB total disc space used, of which ~70GB is the database. You can compress the database and save lots of disc space (50%+) but I haven't gotten around to that yet.
How about buying and owning things?