DIY is the way to go. Buying NAS hardware makes 0 sense imo unless we’re talking (used) SMB / Enterprise stuff. Used computer parts including a mitx board with 4 sats headers and a case that can hold 4 drives is a perfect starter. With drives up to 20 TB being rather affordable per TB these days you can get 40 TB of usable space on a RAID 10. That won’t fit in the $250 budget of course, but you could start with smaller drives or, as I do, forgo RAID for now because all I store is media I can redownload anyway.
The cheapest solution if you want the most basic of starters is an old cheap used NUC with a 3.5" drive slot that you can slap an as big a drive in as you can afford and then go down the more proper DIY NAS build.