You're right, that it doesn't just work as conveniently out-of-the-box on Codeberg. However you do not have to self-host: You merely have to apply to get access to their hosted Woodpecker instance at ci.codeberg.org.

See docs here if you want to try it out: https://docs.codeberg.org/ci/

Onboarding requires a few manual steps, as to prevent the abuse of Codeberg's limited resources. You will need to request access by filling out this form. After submitting, a Codeberg volunteer will review your request and grant you access if your use case is appropriate.

Edit: added quote from docs

Docker-in-docker. Run container agents which can still run docker commands, often by side-car daemon.