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

Since January 7th, 2019, it is possible: unlimited free private repositories on GitHub!
... But for up to three collaborators per private repository.

Nat Friedman just announced it by twitter:

Today(!) we’re thrilled to announce unlimited free private repos for all GitHub users, and a new simplified Enterprise offering:

"New year, new GitHub: Announcing unlimited free private repos and unified Enterprise offering"

For the first time, developers can use GitHub for their private projects with up to three collaborators per repository for free.

Many developers want to use private repos to apply for a job, work on a side project, or try something out in private before releasing it publicly.
Starting today, those scenarios, and many more, are possible on GitHub at no cost.

Public repositories are still free (of course—no changes there) and include unlimited collaborators.