So you got your staging site indexed? Happens to everyone. Here’s a rough guide on fixing it, and suggestions for preventing it.
(thought I’d write this up somewhere)
The fastest way to get the staging site removed from search is remove it via Search Console. For that, you need to verify ownership via Search Console  (ironically, this means you’ll likely have to make it accessible to search engines again, or figure out DNS verification, which isn’t that common but also not that hard).
Every now and then I hear from someone who accidentally got a bunch of email addresses indexed as parameters to some script on their site. There’s sometimes an easy solution to that, which will (temporarily!) take care of it fairly quickly: If there’s a common part of the path that identifies these URLs, use a “directory” URL removal request in Search Console (verify ownership first). For example, you can submit “email.