If you're like me and were staring at Gitea's web site thinking, "dang, how do I run this as a daemon," Gogs (from which Gitea forked) has a nice page on doing this: https://gogs.io/docs/intro/faqs.html
The link shows how to run the app, not how to run it as a daemon. That said, the answer is platform specific, although it would be useful to have some common examples.
[1] https://docs.gitea.io/en-us/install-from-binary/