Synology is more closed down but they make it very easy to use. Note that nas prices don't include the hard drives - that could be $250 alone. Recommend finding one from past generation.

Just went through this myself and settled on DIY build. Someone recommended this video, and I followed it very closely. The built was so easy and less expensive than off the shelf. I installed TrueNAS on it and couldn't be happier.

https://www.youtube.com/watch?v=vjDoQA4C22c

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.

To start with you can grab whatever hardware u have (or buy whatever on ebay) if you are spending money then new hdd and some way to add redundancy. Throw proxmox on it (or TrueNAS), proxmox will serve you in multiple ways. Install cockpit to serve HDDs over samba / nfs.

Video explanation: https://www.youtube.com/watch?v=Hu3t8pcq8O0

Another recommend for build it yourself. I saw deals on 8tb drives at my local micro center and through 2 into an old desktop I no longer used. Been running that with TrueNas scale for a little over a year and its been great. I did recently switch to Ubuntu server after some docker issues that were specific to truenas though. Honestly given the budget you mentioned, a cheaper minipc off amazon and an external drive or 2 is a good place to start off. Its nothing crazy but you'll have something functional to see what you need and want out of a server.

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.

Build it yourself! I bought a used HP desktop from eBay and I couldn't be happier.

I am no expert but I just picked up a qnap ts 462 as my first NAS, really happy with it.

I've got a QNAP. Relatively happy with it, as it's easier to manage than DIY, but it's hard to migrate to a different brand as the RAID implementation is proprietary, so can't just plug the disks in a different brand. First NAS lasted about 7 years until it was too slow, new one is about 5 years and still going strong.

DIY gives you more flexibility, but also more maintenance.

I bought the Aoostar R1 and freaking love this thing. Throw in 2 drives (I bought 2 refurbished white label 10TB drives I got off Bezosland for cheap) and threw Proxmox on it. Serving up my SAMBA shares, running PiHole and Jellyfin, all on separate LXCs. She's a beauty.

I got a cheap 4 bay Terramaster off Amazon for $250 AUD.. they are on special a few times a year including prime day and Black Friday. They are ok. I get about 95MB/s write to a raid 5 array

QNAP. Their current offerings are solid performers.