There is a permission policy that you can set up so that all hired candidates are automatically marked as Private under Configure > Permission Policies > Make all hired candidates private by default
Users with Site Admin permissions who have been granted the optional "Can create and view private candidates" permission can also go to the candidate's profile and mark them as "Private." They'll stay in the system for reporting purposes, but only users with these specific Site Admin permissions will be able to see their profile, interview notes, etc. (For those with a Standard subscription and permissions configuration enabled, job admins could also have this user specific permission)
To mark a candidate as Private, you can check the box Make Private on the sidebar under the candidate's profile.