Proliant N40L. Find one with caddies, swap the CD drive out for a SSD on that connection and put four fat drives in and install Truenas (the BSD based one).Super cheap, low power, fast enough to stream from files stored there, and extremely quiet.
Buildapc
My gut feeling is something like 10+ TB might be a good amount to start. Something that won't fill up quick and that I can put big things in (like a full system image of another computer) without concern.
If you just want a rotational drive array, aren't looking for easy hot-swappability, I'd just start with an ordinary mid-tower case. You can get a 10TB+ rotatational drive today, so all you actually need is space for one 3.5" drive, which is not a lot to ask.
If you're pretty sure that you're gonna stick more 3.5" drives in there, just look for cases that have a bunch of slots for 'em.
If you need more drives than your motherboard has onboard SATA connectors, then you can get a controller card to add more.
If you want to expand later beyond what the tower can store, you can get USB drive arrays.
I personally do not think that used drives are a great idea. There are lots of pieces of computer hardware that can last a very long time, but rotational drives are going to be on the shorter lifetime of hardware that goes. If the company that got rid of them doesn't want them, that's because they don't want to mess with them. I also don't think that you're going to save much if we're just talking about a 10TB server.
The simplest way would be to get an off the shelf NAS (Network Attached Storage) like Qnap or Synology.
And add a cloud backup solution like storj.io, backblaze, etc
Will it cost you money or grief if you lose that data when a drive fails? (note: when.)
If so, then you need redundancy. It can be in the form of regular backups - in which case you need 2x the storage in some form, or some kind of cloud provider - or you could save a bit of space by setting up a RAID5 or RAID6.
Check out https://www.raid-calculator.com/ - for instance, 5x 2TB disks in a RAID5 gets you 8TB storage that will survive the loss of any single drive.
If you can get a bunch of cheap drives (and a board+case that can take them), that can be a decent solution. You can go minimal on all the rest of the specs, boot linux off the cheapest m.2 you can find, and set up an MD device across the hard drives. Keep a copy of mdadm.conf somewhere to easily rebuild the array on a new machine if you ever need to (but it's still simple even without that)
A server is quite literally just a computer. Your desires could be serviced by something as simple as a raspi with a USB attached external HDD.
My very first server many years ago was just a desktop chassis with old scrap parts I had lying around. If you don't have parts on hand you can obviously buy them or if you are unsure about how committed you are to this project just pick up some cheap used system of Craigslist/some online marketplace/eBay/etc.
The above all assumes you are interested or comfortable with setting up the software side of things yourself. A prebuilt NAS device is again nothing more than a computer but among other things will come preloaded with an OS and software catered to fulfilling its role and typically making it highly user friendly to just start rolling with.
If cost savings is key though and you are interested in learning you can also just install Linux on any old box and set it up to do everything and more that a prebuilt NAS can.
I think I may have made myself seem like more of a novice than I intended. I know a server is a computer. I run Linux on multiple devices and I've built multiple daily use/gaming computers. I only meant I haven't had experience with what sort of things to shop for for a file storage server.
The bare minimum is a hard drive and a computer. Unless you have a more specific question I'm not sure how else to answer. I don't think I fully understand what your concerns are.
I don't have any concerns, I guess I'm just curious if there is any sort of advice folks with they had after the first time they did something similar.
Well... I guess advice for such an open ended inquiry is desgin our backup plan first and build around that.
Also decide what level of fault tolerance you want. If you want to be able to survive a disk failure without restoring from a backup then you'll want at least a RAID1 configuration or similar.
If you do so be sure to test out your planned recovery methods before loading up all your data and thoroughly document your process so that when the worst happens all you need to do is follow your own directions.
To your point, I'm running Proxmox on a 5+ year old Dell small form factor desktop with 3 (2.5") drives in it for data. That's all that will phsycially fit.
Works great as a file server, and it syncs the data partition to a NAS and a couple external drives using an offset schedule. It also has a backup running to Backblaze. (The NAS doesn't have the performance of the desktop, and it's even older).
You might want to check if your router has the ability to mount network shares for attached storage.
If you need ~20TB or less of space, I'd suggest that you just add an extra HDD to any computer you already have. You can get that in a single drive for a few hundred dollars US without doing anything particularly special. I can find various 16TB 3.5" HDDs being promoted around $330-ish USD right now with ~30 seconds of looking; you might be able to get better $/TB if you spend longer looking than I did.