Quote by Jurjen: After a user has submitted a story, link or event there seems to be no way for him to edit it without help from the administrator.
Correct. That is not supported.
Quote by Jurjen: Is there a setting or hack or anything to allow logged-in users to edit their own stuff? Or if there isn't, then perhaps a way to allow them to edit any story or event (regardless if it's their own)?
You can add them to the StoryAdmin (LinkAdmin, EventAdmin) group. But then they'll be able to edit most of other people's submissions.
Also see the FAQ:
Setting up a Story Adminbye, Dirk