I want to be able to SSH into my Mac, but it's disabled in System Preferences (Sharing --> Remote Connections, I believe). The administrator disabled this, and I'm looking for a way to re-enable it without an admin password. Is there a way I can re-install SSH to enable it, or something along those lines?
Not easily and you may be running afoul of policy if you are circumventing the access policies of a computer to which you are not supposed to be modifying the security stance.
If you were to run sshd in your user space, you will need to use a port higher than 1024 and also hope that the firewall and network will allow you to accept inbound traffic on the port you chose.
Here is a decent article on how you would do this with admin permissions. It's not a perfect fit for Mac OS - but should give you a feel for what the issues are in your quest to set up sshd.