Use pict-rs, it's what Lemmy uses for image storage, it supports MP4s and GIFs too:
this post was submitted on 28 Jul 2023
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It's honestly not that hard to make one yourself if it's just for private use, with just a little google-fu you could have something barebones but usable within a day or two. That said, if you intend this to be publicly accessible, don't bother unless you have tons of disposable income. Filehosting is expensive af.
I suggest giving a look at the options here, I like PictShare from that list but I also want tags and similar things which it intentionally doesn't have, for your purpose though it should be perfect.
Webm.red
You can upload videos to a danbooru instance you run, as well as pictures/gifs/webm/etc. It is all about tagging, and if you put it behind some auth mechanism (I'm using Cloudflare's application Access portal to use Github as my IDP) you can lock it down to just friends and family.
Kinda overkill but you could join a peertube instance?
It's a completely different use case, gfycat was for posting little inline dumb clips, mostly for memes, forums and similar things, Peertube is downright inadequate for it