Technically - a webserver is all you need. And there're a bunch of FOSS ones. Realistically storage and bandwidth will eat through your wallet if you keep it open to the public.
True. I personally wouldn’t want to host a public instance because of the chance that illegal content could be uploaded and because my isp would take issue with the data usage. I was wondering if someone had done the work in creating something similar. I’ve definitely got the horsepower for a private service share amongst friends and family.
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.
What features are you looking for? As others have said, if you just want somewhere you can store images yourself, you don't even need software aside from a webserver and something to upload with.
But there's also things like user accounts, tagging, browsing/discovery, plus whatever else gfycay does/did.
Anyway, just to actually give you a suggestion, chevereto is used by a friend and it's a lovely user experience (can't tell you about the admin side, though). [edit: This uses folders to organise - no tagging - so it might not meet your needs, which is why I was asking]
Have seen that one and looks like a nice option, but the site and docs have some gaps: can I put metadata/tags on the images in addition to the name for searches later? Also, can I put a description on the folder itself in addition to the description of each image? If both are "yes" I need to set up a small server.
You can add a title and description to images, folders, albums (what we've been calling folders), sub-albums, etc. You can search on those, but it's not a structured thing like tags. I guess you could just store some JSON in there but you might need to get smart with your queries to search. Afraid I have no idea if there's plugins, or even if what I've been using is a recent and/or unmodified codebase.
I think it's more designed for photo uploads, as there's an option to keep exif data, and it automatically makes images of different sizes (including your original, maybe massive upload).
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.
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