If you’re just backing up and not serving this data just get 2-3 4TB drives (new, recertified, or used) and an external dock and test the drive then back it up then test again and check SMART both times. Place one drive with a relative or trusted friend. Connect and power up each of the drives at least once annually, refresh the data with anything new at that time and check the smart stats, consider running at least a quick SMART test to ensure none are mechanically failing then back to being unplugged. Really every 3-6 months would be ideal to power on and check SMART but I wouldn’t pester a relative that often for the external one, 1-2 times a year should be fine for that.

This strategy protects you from cryptolocker malware by not leaving any of them live and accessible.

• What’s the cheapest and most flexible NAS I can make from eBay or local? What kind of processors and what motherboard features?

Cheapest or most flexible, choose one. If you want absolute cheapest but not that flexible you can buy a used office PC, a Thinkcenter or Dell optiplex are the most reliable ones though depending on the model they may accommodate anywhere from 1 to if you’re lucky 4 (though commonly only 2) drives via that many SATA ports (often half the SATA ports are 1.0/2.0 for DVD drives so you may not get full speed). Finding space inside them for more than 1 drive could also be a problem depending on form factor but mid-tower models often have room for 2 with space for a third lying on the case itself if you really want to push it.

Most flexible I suppose someone else’s old NAS build, a used case with room for at least 4 3.5" drives gives you a little room to expand.

• What separate guides should I follow to source the drives? What RAID?

You don’t need RAID, it’s not a back-up solution, RAID is for high data availability and integrity. If you really want to you can set-up a RAID 1 I suppose though know this means you’d require at minimum 4 disks for your data and one copy and 6 disks for two copies.

As to sourcing the drives, there are various companies, server parts deals is one that’s well known and decent though their presently available sizes may be larger than what you’re after. No matter whether the drive is brand new, recertified or bought used on ebay the recommendation is test, test, test. Even new drives can be bad. Run a full SMART test at least once, check the SMART data and make sure there are no failure indicators. If you want to be really thorough I’d suggest checking the SMART data when you get it, noting anything concerning, running an extended/full SMART test then after that finishes formatting the drive but unchecking quick format and doing a slower format option that writes zeros across the drive, then filling with your data, then doing another full/extended SMART test and again checking the SMART values before putting it away. Re-test and check SMART at least annually if you’re keeping the drives cold.

• What backup style should I follow? How many cold copies? How do I even handle the event of a fire?

At least two copies, ideally three, at least one copy off-site for things such as fire. If you don’t have a relative, friend, or workplace where you can stash an off-site copy your option would be basically cloud storage back-up which for 4TB wouldn’t be too costly (backblaze personal would allow this much IF you keep one copy connected to a computer that has their app and is turned on at least monthly and they’re $100 a year though note they will delete your data if you go more than 30 rolling days without syncing so if there is a disaster you have a limited time to either get another drive and download it again or contact them and pay to have a copy shipped to you before it’s deleted).

You could also I suppose invest in a fireproof safe though that doesn’t protect against burglary where they steal your safe thinking it has valuables in it. You really need a copy off-site. Other options would be a bank safe deposit box though probably more costly.

One way to get friends to help is to buy more storage space than you need, say two 8TB drives and you offer to back-up a copy of their stuff at your house so you have a copy of their stuff+yours at your house and they have the same copy at theirs. Though you could also use separate drives.

Most are redundant video files that are in old encodings or not encoded at all

All re-encoding unless it’s from lossless to lossless induces degradation. For archival purposes I’d suggest against re-encoding unless it’s to another lossless format or unless they’re in a lossless format or very high bitrate (>20MBps video for SD or 1080p HD) and you’re keeping a high bitrate in the new encoding. Also avoid hardware encoding, it’s faster but introduces more degradation and is less precise than software encoding. Removing duplicates is another matter.