Here are my personal thoughts on it:
Why deal with this at all?
Because some unanswered questions are difficult but interesting (at least to some people), some are irrelevant, some are really already answered in the comments but so simple that no one has posted an answer ...
The point is to make the important ones a bit more visible, and get rid of the rest. This refers to old unanswered questions, not ones posted only a couple of days ago.
How to deal with it?
I have the impression that there are actually lots of questions that are not so difficult to answer, but it's a lot of work to implement the solution. Many have good enough answers in the comments already, but people here tend not to post in the answer format when they didn't work out the full solution and don't have code for it. Sometimes this is really simple to write up the code, but it's also boring and not worth anyone's (but the OPs) time. I think often (but not always) it's enough to just give enough information to solve the problem. It's not necessary to write all the code for the OP... When someone already made a comment which practically answers the questions but doesn't have the code, why not just re-post the comment as a community wiki answer?
Similarly, when the answer to a question is "this is a bug", it's better to get the question out of the unanswered list by posting a community wiki answer.
How to organize dealing with it
Once we come up with a few guidelines on how to deal with various types of questions, we could organise bi-monthly cleanup days/weekends.
I think what we need to significantly reduce the unanswered count is to set some guidelines on how to deal with certain kinds of old unanswered questions, and make these guidelines "official". Then people won't be afraid that they'd break some unwritten etiquette rule when thinking of posting a very simple answer.
I'm going to post a set of proposed guidelines as a community wiki answer below (so everyone can edit it). Please edit it or post alternative suggestions as another answer. The aim is to end up with a set of guidelines everyone agrees upon.