It happened to me couple of times: I write answer for unanswered question, click submit, and now I see 2 answer, mine and other, posted just a tiny bit earlier. They are duplicates, or I find the other one more comprehensive. So I want to delete mine - it's a duplicate, or should be edited to be full and then it would be a duplicate. And if I'll ever came up with another answer, it will be totally unrelated to this one. So, is there a way to delete it permanently, make it gone and don't waste database space? And get rid of seeing it every time I stumble around that question?
As kiamlaluno said, the site doesn't give an option to hide deleted answers.
However there is a user script that will hide deleted answers for you; once you hit 10k it should be easy enough to tweak that only to hide your own deleted posts (until then it won't make a difference).
Posts are always soft-deleted, including questions. Deleted answers are visible only to the user who wrote them and 10K users.
There isn't a way to hard-delete them, or make them invisible.
The first is not desired, since moderating user's actions require to be able to see what that user posted; if a user keep posting as answers something that doesn't answer the question (e.g. another question, a post promoting something), and hard-deletes it after receiving two down-votes, it would be impossible for the moderators to catch that behavior. (Moderators see the posts users created from their user profile because posts are soft-deleted.)
The second would be possible if there would be a setting for the users to select, but it seems Stack Exchange doesn't like much user settings. (In fact, there are only two user settings.)
Maybe having a setting for deleted posts would be fine, but that is a feature request to ask on Meta Stack Overflow.