How secure are private Git repositories?

> Would you use them to version sensitive data?

If you are using a self-hosted version of Github or preferably GitLab then yes. However, If you're on GitHub or GitLab's cloud version then it's not secure and you have zero control, even if its private.

> Would Gitlab/Github have access to the underlying content/history?

Who knows. But the first answer tells you that you will have more control in a self-hosted environment over a cloud based version and I wouldn't risk putting sensitive data there unless I have complete control with a self-hosted open-source version (GitLab).

> ...though how about medical/income/taxation documents or information?

Well that's very sensitive data equivalent to bank-level information, which can be used as a reason for others to determine your job, insurance or loan choices. Thus, should be treated as sensitive too.

I have often wondered this myself. Considering if github was ever compromised, could an attacker modify your source code without your knowledge. Seems like a holy grail of attacks.

What would be interesting if there was some at-rest encryption and maybe some audit functionality.