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I don't have an answer since I don't know anything about NAS's, but I was considering doing the same thing to stream video using Jellyfin. I was thinking of building something using my old 3570K based computer, but then I was also thinking it might be more power efficient to get a cheap Raspberry Pi or one of the many alternatives that are now available.

Edit: The Raspberry Pi 5 has PCIe, so I was thinking that I could add some kind of expansion board that adds SATA ports, but again, I don't know enough about how any of this works to know if that would even be a good idea. It would definitely be a lot easier and simpler to just use an old computer, but I was thinking it might use a decent amount of power running 24/7. Maybe it won't use as much as I think since it won't need a separate GPU and I could probably undervolt it a little.

I don't believe you'll get anything decent for under $100 (if at all).

For streaming video (transcoding or not), check what DLNA is. I use Serviio, which is freemium. I never needed the premium side so only free covers my needs to share anything in my computer, to watch it in the TV. Allows transcoding and burnt-in subtitles, tho I never used them. Good personalization of what and how you see things in the TV.

Also, is it better than homemade?

Nope, they usually come with their own proprietary OS/Firmware and their hardware is difficult (if not impossible) to upgrade. Once they reach their EOL they usually become a paperweight (and a dangerous one since you don't receive security updates either). I've had a few low budget NAS (Conceptronic and Xtreamer e-TRAYz) and I wouldn't go back to using those.

If you don't mind the size and noise, an old PC, Unraid (or FreeNAS, but I like Unraid more because of the flexibility of their "RAID" implementation) and a bunch of old hard disks you might have around is the way to go. You can easily escalate it adding new disks or replacing old ones with bigger ones and Unraid has tons of features thanks to its plugins system and docker support. You might be able to find a second hand PC for less than $100 that is very capable of working as a NAS.

Another solution I like to use is a system like HP's ProLiant MicroServer and, again, Unraid. Latest generations are not really cheap (not under $100, that's for sure) but they're also very versatile, you can run anything you want on them, and their hardware can be upgraded easily. You might be able to find old gens (N36L/N40L) at good prices, though.

AvidWriter suggested using a Raspberry Pi setup, which is fine, but it has a few drawbacks: you have to use external storage (since it will be bulky why not take the old pc/small PC route), the USB external connection is not so fast and/or reliable as physically connected hard disks and performance can be a bit meh as soon as you try to get more juice out of it. The hardware is non-upgradeable, too. But yeah, it's a decent option when you're on a budget.

Another way would be using a Zyxel NAS542. That would be a 4 bay solution. You can install custom OS via sdcard. Things like Debian or openmediavault should work just fine.

hot take

a decent router, a powered usb hub, and some ssd's satisfies most normal human beings requirements

are you backing up Tb of rendering videos daily? are you trying to play games off it? is it supposed to satisfy the needs of a dozen people?
if not, ...

i HAVE A TP-LINK AX1800 W A 1GIG SSD ON USB 2.0 IIRC. IT CAN SERVE VIDEOS TO 3 AND WIFI with only the rarest buffering. It supports daily backups too. Spend another 20 and you get way more bells and whistles but for my money this is way better than rolling the dice with some buffalo wal mart nas

Anything less than a couple hundred bucks or not from a known brand is going to be a bad time. You won't match the capabilities of a self built system until you're up $1000+. There just isn't much competition in the consumer NAS space.

IMHO the ideal solution is using truenas, formerly freenas. https://www.truenas.com/truenas-core/
Hardware wise you are right the downside of an old PC is power usage. Get a mini industrial PC and a USB3 JBOD enclosure, $300-$400 would get a badass setup and can do double duty as a router (pfsense, etc.).
Something like:
https://www.aliexpress.us/item/3256804996123450.html?
and
https://www.aliexpress.us/item/3256804757382284.html?

If you are running it 24/7 the power savings will make up the price in a year or two depending on your electricity cost.