If you wanna stay in-fediverse, you could post the pics on a Pixelfed instance and use that as the host. Just post the pic there, copy the direct URL to it, put it in the URL bar in a post here and it will show up as a normal picture instead of a link. Also Pixelfed allows unlisted pics, like if it's stuff that would clutter up your profile.
Question from a newbie: if you happen to choose an unlucky server on pixelfed, and it gets closed after a while, all the pictures will just vanish, right?
I mean, nothing to worry about if you are hosting memes or other "need to be fresh" content, but if you are looking for something that can be relevant even in the future, maybe that's a bit risky.
You're probably right, yeah! In that case probably one of the other commenters' suggestions would be a better choice, I think. I'm hoping the pixelfed stuff will catch on enough that at least some instances will be more likely to stick around, but it's anyone's guess if that happens.
If the image is from anywhere else on the internet, you can link it from there. I found myself downloading things out of habit. Idk about image hosting sites, they have a pretty unsustainable business model, so I doubt any particular one will be around in a few years.