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)