Is it possible to access a Mac via SSH without administrator access?
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.

Also, netcat.