There's people running old raspberry pis with USB hard drives.

It'll run on just about anything.

Though, you'll only be able to stream original quality, no on the fly quality changes for low speed connections and such.

NUCs (specifically Intel 8th?-gen or later) are pretty much ideal for serving Jellyfin because the Intel integrated graphics can do video transcoding and the software is actually not very demanding otherwise, so the low-power CPUs are fine.

If you were buying hardware specifically for Jellyfin (i.e. didn't want to cobble together something used), I'd suggest an N100 or N150-based NAS mini-PC like this: https://www.bee-link.com/products/beelink-me-mini-n150

I have my jellyfin running on ubuntu server on a 12 year old PC with a bunch of salvaged HDDs added in, plugged into my standard home internet. I have like 10 people I share it with. So far it's not made a noticeable dent in either the internet usage or the electric.

You absolutely can. Currently running mine off of my desktop, but it's overkill and have a few coworkers that used nucs/cheap mini PCs to setup jellyfin. Biggest thing is to make sure you have enough storage to hold all of your media, then you're fine.

Mine runs on my desktop that I built in 2016. So yes. I also tested it on a Lenovo tiny (similar to a NUC) that I'm using as a self host "server" and it seemed fine but I didn't try any heavy transcoding yet.

You can definitely run it on those sorts of machines. You'll want plenty of storage, but apart from thst it's not too demanding if you don't load it up with very high res videos.

Running on a 2016 intel processor with an old Nvidia 2070. I can watch about 3 streams on different devices at the same time.

I could do something simple like a NUC or equivalent and a tiny NAS

Literally me rn. A tiny second-hand Dell Optiplex with a 8th gen i3 and a 2TB SSD.
Maybe not the most future thinking solution since it can only fit 2.5" HDDs and NVMe's, which are both more expensive than conventional big fat HDDs, but hey, works great.
If you can find one for cheap locally and get a decent deal on a compatible drive you're set. You could stretch a Terabyte or two for a while as long as you're not trying to host Jellyfin for too many people (and are OK deleting watched Shows/Movies when you start running low)

You can probably run it off your existing hardware just using an external drive(s).

i mean i'm only using it for me and my family but i just run it on my desktop linux (and before i switched i was running it on my desktop windows with scoop install jellyfin)

My buddy had success running it off a pi.

But he had to encode everything ahead of time as h264.

Your biggest issue self hosting JF will likely be port forwarding and the infrastructure required so external users can connect via WAN. Plex abstracts a lot of this away since you just have to open the port and Plex will access your library and broker the data between your server and the client. This now requires the server owner have a Plex Pass. The easiest way to host JF is allowing users LAN access via WireGuard (pivpn) but then you start hitting limitations to which devices they can install WireGuard on (smart tvs). I use Plex and JF, Plex for external users since I bought the lifetime pass 10 years ago, and JF as a backup as the enshittification of Plex continues full steam.

But then won't your rack look a bit empty?

I just run it on my gaming tower in a container

Mine runs on a mini PC(NUC) hooked up to a hard drive array for the storage. So it's basically a tiny PC and another box full of hard drives(not required, but you'll need space somehow...). Pc was around $250.

Very easy and you don't need to set up an actual "server rack". Hell, you can use an old laptop.

Also, keep in mind you can hook the miniPC up to your TV or another PCs monitor(assuming you have extra plugs). You don't need a dedicated monitor for this. Mine uses the same monitor as my gaming PC on a different input. It basically lives on my keyboard tray tucked away running.

an entire data center

What are you on about?

They're probably saying that they run the self hosted jellyfin app and stream their own media. That's what I do, and have been doing for like a decade because netflix and other shitty streaming companies can eat a bag of dicks (except Dropout, they're real Gs) from the number of times I've gone to watch something on my watch later list only to find out they didn't get the license this quarter so 🖕to my movie night

cough hoist the flag, and watch netflix shows on jellyfin cough