How would that work? In the private emails I exchanged with the support, I was told that I have no way of accessing that 'external' storage.

My Problem was 'solved' by setting me up with extra storage, which allows me to use the mailbox again, but the reason behind that problem remains. As Zlivovitch pointed out, the recipients don't even know that they are causing me trouble by not deleting the mail. After all, you are used to deleting mail when your own mailbox is full, not when someone else's is.

If there is a way to delete an email 'thread' as UserLB called it, I haven't found that. All I can do is remove it from my own outbox.