I thought I remembered having seen some discussion of this. It's in the question vote to answer?, though that's not a very search friendly title so no surprise it took me some time to find it.
Have a read through dmckee's answer to this question, and in particular follow the links to the mother meta. Although there has never been a definitive statement by the Stack Exchange team they have not implemented a community accept feature despite numerous highly upvoted pleas for one. This suggests it's not a feature they regard as a priority. Maybe it will be implemented one day, though I wouldn't hold your breath.
In the mean time note that answering an old question will bump the question up the rankings, though I don't know how big an effect this is. Answers to old questions also show in the review queue for users with enough rep, and I do upvote any such answers I consider good (though sadly many answers to old questions are, well, rubbish). The upvotes boost the question rankings and further enhance the visibility of your answer.
There is also a Necromancer badge for answering a question more than 60 days old, and 475 of these badges have been awarded. So not only is there considerable activity in the answering of old questions, but since award of the badge requires at least 5 upvotes there is clearly considerable rewards to be had by doing so. On a side note I note that both David and myself, arch villains though we may be, both have the Necromancer badge.
More broadly, I think we need to distinguish between questions that have no answer and questions that have answers but no accepted answer. The latter normally means the OP is AWOL or just doesn't care, and in these cases there is little we can do. While the OCD side of me deplores perfectly good answers being left unaccepted this doesn't lessen the value of the site to the physics community. Accepting an answer doesn't mean it's right - it just means the OP thought that was the most helpful answer. A more reliable guide is the number of upvotes the answer has attracted.
If a question has no answers that probably means either the question is really hard or that it's really uninteresting. Anyone giving a good answer to an old hard question is likely to find themselves amply rewarded (as the number of Necromancer badges shows). Conversely, if the question is uninteresting it's likely your answer will be ignored.