They only allow private repos as an exception, and only insofar as they're ancillary to open source projects.

From their FAQs[0]:

> Codeberg's mission is to promote free/libre software. Keeping software private is obviously not our primary use case, but we acknowledge that private repositories are useful or necessary at times.

0. https://docs.codeberg.org/getting-started/faq/

I'm more than happy with https://codefloe.com for private repos. The service is blazing fast and the maintainer does a great job keeping it up-to-date with recent Forgejo releases.