How can I get to files on my UNET webspace from a department Linux machine?

You can use SCP or SFTP. LTS changes their shell servers from time to time. As of this writing, a list of their available shell servers can be found at http://lts.brandeis.edu/techresources/accounts/unet/serversfilestorage.html, which states that you should use limpet.unet.brandeis.edu or starfish.unet.brandeis.edu for the hostname/server. For the folder use /usr/users/<USERNAME>, for the username use your UNET ID.