NAS and plex are not the same thing.
NAS is just a storage means, can have a variety of protocols by which it is accessed but that is what NAS extends to.
At the same time plenty do have other options, including plex servers and torrent clients (you usually upload a torrent file, or possibly magnet link these days on a web page it hosts, and it sorts the rest for you, whether you stuff that down a VPN or need to sign into a private tracker then getting a bit more tricky).
https://www.truenas.com/apps/ but we will return to that shortly.
For my money there are three types of NAS.
The cheap junk they sell to consumers to have a couple of drive bays possibly in a RAID and serving files via windows file sharing, FTP and maybe something else. Fairly narrow in their use but OK enough for a small office to share files* when the main PC is not on/not having a main PC, or some nerds to use in their own house knowing the limitations.
The expensive nonsense they sell to businesses with all manner of extras you can pay for to do real things, and might well come with enough drive bays to do something real.
Those that get an old PC, a copy of FreeNAS, or truenas core as it is known these days
https://www.truenas.com/truenas-core/ , and get something real going on. Some also do fun things with raspberry pis here but that is a more specialist thing more in line with the cheap junk albeit the with the ability to run a few applications on top of it. I already linked
https://www.truenas.com/apps/ but have a look and see what is available.
Some will go the other way and run your Plex server setup as the storage platform itself (though it seems standalone ones are gone and instead it wants to be an application on a more general server) before manually setting up everything else -- many motherboards will have RAID if you want that and it is not hard to set up SMB (windows file sharing), plex itself, FTP and whatever else.
Hard drives in NAS world are interesting thing. You might have some already, however you might also consider some of the drives designed to be run 24/7 (various companies doing it for NAS and CCTV usages). Most consumer drives will hold up well enough though.
I mentioned RAID above. If you are not familiar with RAID then it is a way of combining discs together. There are various types ranging from RAID 0 which stitches two drives together theoretically making them faster but any one of those drives failing and the whole thing goes, to RAID 1 which is a mirror of the drive (one failing does not matter as the other is an identical backup, obviously though you have half the storage if they are the same size drives), to various combinations of that (RAID0+1 needing 4 drives but theoretically being the best of both worlds), and ways of having drives hold redundancy data so one drive (of several) can fail but recovery data is held on the others such that over the next however long (potentially hours if it is terabytes) it can restore it.
https://phoenixnap.com/kb/raid-levels-and-types
Any NAS worth anything will offer this, and more, as an option. RAID is the absolute basics of big boy hard drive data storage guy skills but probably all you need here.
CPUs capable of transcoding are more or less anything that is a PC really. Time was it was a CPU killer but today then eh. Only thing I would note is if you want to use the GPU encoders that might take a real GPU. Also are you sure it needs to transcode? Most things should be able to decode the baseline video, though I suppose if you want to serve up a 360p version of a 4k blu ray rip to a phone then why not. A raspberry pi with XBMC on every TV that needs one is potentially a game changer for you in this.
RAM per user if this old PC was vaguely current in the last 7 years should not be an issue. What I am more here to say though is be very careful when speccing things out like this -- I know you want to build to the worst case scenario but it gets very expensive for what might only be a few hours per day. That said if this is a media box then everybody hammering it from end of work until midnight (any vaguely good torrent collection is going to be better than 5 streaming services combined) when they go to bed is a more plausible scenario and I doubt you can do anything cute**.
Are ethernet ports that rare these days in your searching? All but the cheapest of cheap junk tended to include one from what I saw.
Likewise all should feature some kind of management interface, usually a web server embedded within the NAS like you might have for a wireless printer but some will also give you SSH or something if you want that instead. If you are running a server machine with plex on top and whatever storage it has underneath all that then you do also have the normal selection of remote access tools like teamviewer, anydesk and whatever else the kids are using this week.
Remotely outside your house will likely involve fiddling with the router (some things can bypass it these days, quite thankful for that one with everybody wanting CCTV, but I would not lean too much into this. Instead some kind of dynamic DNS service (assuming you don't have a static IP for your house) and setting ports on your firewall. Whether you go one step further and have your users ssh into your server to do anything is another matter; a while back you mentioned being blocked by work so this would help with that as well as being generally more secure.
On BIOS (assuming you are doing the full PC route as opposed to "device") then most these days will have an option like restore last state after power loss. I will note though that if you are using a RAID setup it might take a while to come back online if it has to scan and possibly rebuild something.
It is also at this point I would suggest getting a UPS (uninterruptable power supply) such that it can keep things running if it is just a 30 minute outage or something (plug your router into it as a well) but more importantly it can initiate a proper shut down to avoid having to do said rebuild.
*I have a client who has been in business for 15+ years at this point. Ignoring his photos directory and emails (which are on another system anyway) then every spreadsheet, document and more I moved from computer to computer over the years barely totals 50 gigs and he does multiple every day, as do his people. This is then very different from what you want where a terabyte might well be a TV show that ran for a few series in super high quality (4 gigs a pop, 23 episodes a series, 10 series say and you are getting up there).
**variations of this story are told to new IT peeps wherever you go. So there was this large business and every day after lunch their login servers would get absolutely nailed and crash sometimes, work fine the rest of the time though. Obviously everybody having their 30 minutes lunch or whatever would come back at the same time and try to log in. Manglement then specs a supercomputer to handle it, until the far more obvious solution of just stagger lunches by department so everybody does not come back at once was pointed out. Harder here, that is unless you want to build a couple of these, put them in the houses of your friends/family and have access to them such that when you add things to the list it automatically replicates it on their local setups over the next few hours. Which has the added bonus that if yours gets hit by lightning or something then you can just copy it all back off one of their machines.
