How do I set up a private Git repository on GitHub? Is it even possible?

Bitbucket - Their plans seem to be the best. They give you way more than GitHub do for free accounts - in fact, I'm still only using the free plan - no need to sign up to the paid ones; plus the interface is almost identical to GitHub.

A repository on Bitbucket can have up to five private users with unlimited public or private repositories - the only thing you seem to be paying for with the paid accounts are more users to access your private repositories.

On January 7th 2019, GitHub announced free and unlimited private repositories for all GitHub users, paying or not. When creating a new repository, you can simply select the Private option.