Would there be any potential problem of hosting public and/or private (vpn) services in a school office?

Using public property for private usage likely fall on the bad side of acceptable use policies.
- It is my own device, but yes utilities and security is their own.
  
  If you are so sure of your indemnity because it's "your device", why are you asking on Lemmy?

What specifically are you attempting to achieve, because right now, what little you have shared sends up red flags and rings the alarm bells .. loudly.
- I have a server in my school office. I currently only use it to backup important files. I am asking if running public or private containers on it would be safe and acceptable.
  
  If authorized by the school IT department and policy, yes. Ask them, not us.

It's likely illegal. The administration would call it theft of service because it's not authorized and they wouldn't be wrong. I also don't see why you would want to do it. You're giving the IT department at your school complete access to your web history.

I've worked for a university before and it was very common for staff to remote into their systems from home -- usually with SSH for CS types or Remote Desktop/Team Viewer/etc. for less computer-focused folks. (The former usually didn't have much issue -- the folks using the latter mechanisms got compromised a number of times... -.-) There was also a campus provided VPN that was required to access certain systems with instructions to students and staff on how to use it, but other systems just got public IP addresses.
If what you're doing is related to your work and campus IT doesn't object, you're probably fine to do it. I've run various kinds of websites and web apps for colleagues to collaborate on research projects. Being able to do things like that is kind of the point of the internet.
Having seen a number of students, uh, push the limits and find the boundaries of acceptability the hard way though... I'd strongly advise you not to install cryptominers, run TOR exit nodes, or torrent TV shows/movies/etc. That kind of thing tends to get your systems in hot water with IT or other parts of the bureaucracy...
- Thanks for sharing your experience!
  That’s a great point, that is exactly the point, haha. The public sites are related to the library and my research so it seems that would be allowed.
  Yeah… that definitely would be too far. I’m even careful what I backup (no torrents only work) to my school machine over rsync ssh just to follow the rules.
  What was the most ridiculous or funny boundary push you saw?
  
  What was the most ridiculous or funny boundary push you saw?
  Trolling someone by attaching a camera to the ceiling right above their keyboard. I've been paranoid since I saw that stunt pulled... They got their point across about physical security though.

It's definitely not legal, especially if your school is funded by the public. That "free internet and power" is paid by someone, and if it's the public, it's kind of a dick move.
They can't see what's in your ssh or VPN tunnels necessarily, but they can usually see where the packets are originating from and going to. So if you're say, accessing it from home directly to the server via VPN or SSH, if you're not doing so using a full VPN service like Mull, they'll be able to see the origin IP of your SSH or VPN handshakes, and thus your home IP.
- It's definitely not legal, especially if your school is funded by the public. That "free internet and power" is paid by someone, and if it's the public, it's kind of a dick move.
  This is assuming a lot and I think incorrect. I use the power and internet everyday as part of my job. Doing research and that which is related falls under that. And no it’s not funded by the public… but your point is well taken!
  Thanks for sharing info on ssh. That’s helpful. If I’m sshing via Tailscale I wonder what IP they see? The Tailscale 100. one?