I want to use another linode to store database backups.

My application will write the backup files to any (local) directory that I tell it - so long as it is accessible to the file system

how would I go about doing this?

A simple to set up option would be sshfs. All you need is an ssh server running on the system with the storage, and the sshfs program on the system that will be generating the backups.

My understanding is that it does not perform as well as a "real" network file system, but if you are just doing an occasional backup and restore rather than a bunch of random reads and writes that shouldn't really be a problem. If that doesn't cut it, this article covers NFS, one of the first and still prevalent network file systems. It also mentions a few alternatives like Ceph, GlusterFS, and OpenAFS. This discussion thread contains some opinions of Linode users.

Thank you!