Retag and push to a local registry. Lots of options out there for setting one up.
Honestly, you already have the image locally if you've pulled it. You don't really need to run a registry unless you're dead set on it. You can also flatten and export containers for backup if you really want.
Two good points here OP. Type docker image ls to see all the images you currently have locally - you'll possibly be surprised how many. All the ones tagged <none> are old versions.
If you're already using github, it includes an package repository you could push retagged images to, or for more self-hosty, a local instance of Forgejo would be a good option.
I'd also like to add that you can save an image to a local file using docker image save and load them back using docker image load. So, along with the options mentioned above, you have plenty of options to backup images for offline use.
Well the question is about a container disappearing from a public registry, in which case nothing would happen if it's already pulled locally. Figuring where to go from there is the other half of that problem.
But anyway, it's unlikely docker.io or quay.io or ghcr.io will go completely offline. If anything they might experience a DDoS, in which case I imagine they have competent devops employees who would ensure they become functional again within a matter of hours.