Newb: *Reasonably* fast file server for ~£/€500 all in...?

your son may have a variety of desires that are conflicting, so he may need to choose.

how much total space do you think you need to have for backups? And what kind of networking infrastructure exists (or are you willing to install?) wifi only? ethernet? how fast?

For portability, maybe consider and external drive. like thunderbolt or the fastest USB available. Plug it in and do the backups. That’s the most portable. You could then work out an external storage to migrate it to a larger server later (I think OMV has an external storage plugin? haven’t used it).

But if you’re building a server, with multiple hard drives, and power supplies, and networking, portability tends to go out the window. The closet thing might be a mini ITX nas like this other serverbuildser is planning:
New user in site to built NAS - General / Introductions - serverbuilds.net Forums

If your networking is wifi or gigabit ethernet only (and borderline 2.5gb ethernet) you can just backup directly to hard drives, they can keep up. If you are talking 5gb or faster then you will need SSDs to absorb the writes if you want to go as fast as the network allows. a decent SATA SSD can keep up with 5gb, 10gb will require nvme or SAS (or raid 0) to keep up.

mergerfs can be configured as a “write cache”, and when with cron scripts to push data to the spinners. I eventually got it set up successfully when I had a 10gb connection between my workstation and server. (now it’s just wifi so it doesn’t matter).