This is not something which anyone has any control over but you, I'm afraid. Please see the MSE post on the subject.
Automatic bans never expire or "time out". This means that you cannot simply wait for a certain amount of time. If you do not take action, you will never be allowed to freely post again. The only way for the ban to be lifted is by contributing positively to the site in other ways.
Moderators cannot lift the ban.
Before you do anything else, fix your existing posts! As noted above, downvotes cast by the rest of the community factor into the ban - so the single best thing you can do to get it lifted is to address any objections raised by others. Were your past questions unclear? Did they fail to show any effort on your part? Poorly worded, titled, formatted, and overly long or short? Then fix them!
Note the emphasis on fixing. Do not delete your posts. As explained above, negative marks on deleted posts do still count towards the question ban, and the fact that they are deleted can also count against you (with the exception of questions that are 30 days or older when deleted). Deleting your posts does not help to lift the ban. Only fixing does! You can see a list of your own deleted questions and answers.
You have written 49 answers, of which 38 have been deleted. Of those 38, 27 have a net-negative score. Some of those you will not be able to fix and undelete. But the only way of lifting the ban will be to fix and undelete those you can. If you deleted any yourself, you should be able to undelete them yourself once they have been improved; others you will have to rely on the goodwill of the community to undelete.
Editing a post allows downvoters to reverse their downvote, if the edit warrants that. So converting your existing answers into good answers may well allow the ban to be lifted.
When deciding which posts to edit, check that the question is not deleted — you can't have an answer to a deleted question — and then favour those which you yourself removed because you should be able to undelete them yourself. Note that restoring an answer will bump it on the front page of the site, which will attract attention and that could mean more downvotes if it's not good enough.
TL;DR> The ban is imposed because the community found your answers not to be good answers, and the system protects the site by banning people who don't write good answers. You can't get the ban lifted to write more answers without providing good answers, so the only way of writing good answers is to improve what you already have written.