Are private git repositories safe?

A git repository is just files. So you're asking "Are private files safe?" To which the answer is "you're asking the wrong question".

A git repository is exactly as safe as the place that it storing it for you. No more, no less. If it's GitHub, then it's exactly as safe as GitHub is, And before you ask how safe GitHub is: nobody knows the answer but them.

Same story for BitBucket, Gitorius, Dropbox, Google Apps, Microsoft OneDrive and literally everywhere else you can store files (including your Git repo): Nobody can tell you how safe they are because nobody knows but the vendor. And the vendor always says they're safe.

If you're paranoid, keep your files on your own hard drive. In a mattress. Buried behind the shed.

Technically, the vendors "Github, GitLab, Bitbucket etc." have access to all your source in "Your Repo" because it's "Their server" and "Their SaaS".

Their Server:

The vendor most likely has root access to their servers where your private or public repositories are stored in.

SaaS

It's the vendor's software that you subscribed for. They just have given it a different meaning. Assume your facebook account.