My NAS is an 8th gen i7 Dell Optiplex I got on ebay for like $175. Running TrueNAS CORE. With a shucked 18tb WD Easystore that was a black Friday deal at $200. Watch https://shucks.top/ for good deals on storage. My rig is a little more than you want to spend, but you wouldn’t need the 18tb drive to start with. And you could save a bit more if you went with an i5 instead.
I’m on Proton. It’s easy to set up on Ubuntu and Ubuntu like systems. I’m on Mint. Haven’t tried it on other distros.