The budget is way too low imo.
You could repurpose an old workstation, bought dirt cheap on eBay if you are lucky, but even then you’ll have to get yourself an HDD, maybe multiple of them if you want to have data redundancy.
For anything new your best bet is a 2 bay ready made NAS, but you’ll have to invest around 300€ for the cheapest one.
You could run a WebDAV server, like Nextcloud.
On windows it supports thin sync (meaning that it keep a reference to the file instead of the whole file), on Linux not yet, as it is still in alpha (but you can just connect it as a remote disk and be done with it. That’s how I do with mines).
If you don’t want the whole Nextcloud, there are standalone cli WebDAV servers.