I don’t use pocket like services myself but when I’m looking for alternatives I like to start at alternativeto.net. It’s a great tool for questions like this.
Good shout, that site is a real lifesaver! I'd also like to throw SaaSHub and LibHunt into the mix, the latter is a bit more niche since it only does open source but it does have apps like Omnivore, one of the alternatives I suggested in another comment.
Next cloud is very heavy for just this but the book marks plugin and floccus browser plugins do what I needed for pocket. I used it as a shared bookmark service and didn't really leverage the offline content.
I just finished setting up Wallabag and it's great. And it really shines with Miniflux RSS with it's dedicated "save to Wallabag" button. I love it. Omnivore is great too but I wanted self host so if they ever have the full version self host able, I'll check it out again.
Not exactly like Pocket, but I started using Linkding for basically a bookmark service. It has a browser extension and also an option to mark links as unread so that you can find them easier later.
It works as a read later type of thing, but doesn't have offline sync etc.
Everytime I've tried such a thing the lack of proper mobile support stopped me. Most of my usage is mobile so having to copy paste urls to a webpage and save them doesn't really work for me.
If you're open to not self-hosting it, but the software still being open-source, Wallabag also has a hosted version: app.wallabag.it
I really recommend it, works great. Might consider hosting it myself somewhere in the future, but i also like that i can support the creators this way.
I've been using Omnivore for a good while now, not self hosting for now but there are instructions. The service itself works great and the UI/UX is really close to Pocket as far as I remember, although the Android client is still pretty barebones.
I haven't used it myself but Wallabag also seems to be a popular option.