Fundamentally, the repository you have on GitHub is the same thing as the repository you have on your computer when you clone it. Pulling and pushing are shorthands for synchronizing commits between the two repositories, but you could also synchronize them directly with somebody else who cloned the repository. As somebody mentioned, you can also just host the same repository on two servers, and push to both of them.
The issue is that git doesn't include convenient features like issues, pull requests, CI, wikis, etc., and by extensions, those aren't included in your local repository, so if GitHub takes them down, you don't have a copy.
An extra fun fact is that git can be considered a blockchain. It's a distributed ledger of immutable commits, each one representing a change in state relative to the previous one. Everybody who clones a repository gets a copy of its entire history and fast forwards through the changes to calculate the current state.
Git exposes a lot of internals through odd commands, so I suspect you could manage synchronization by sending changes over email or something.
Bonus fun fact: there's a
git bundlecommand that "dumps" the repository into a single file, that can be interacted with as a remote. So if you're ever working with a local repository and want to put it on a server over ssh or something like that, you can just create a bundle, scp it over, and clone from that on the server.