We already have precisely this feature. In fact, we do one better—we automatically migrate the closed question to a more appropriate site on the Stack Exchange network. The asker doesn't even have to lift a finger. When they try to visit their question, they are automatically redirected to their migrated question on the new site.
For example, this question was recently asked on Stack Overflow and was migrated to Server Fault.
Anyone with close vote privileges (users with 3k+ reputation and moderators) can vote to migrate questions. This functionality is built into the close vote dialog. Likewise, anyone who can flag questions can choose the "it doesn't belong here, or is a duplicate" flag reason and suggest a more appropriate destination site.
Of course, what you've probably seen are a number of questions that got closed as simply "off topic" and were not migrated somewhere else. That's because either:
The question was just a bad question. It was either poorly asked, not constructive, lacked sufficient context and details, or something else. And we don't migrate crappy questions because that would be like dumping our garbage on our neighbor's lawn—not very nice. Those questions get simply closed as "off topic" without the migration part.
The question, no matter how it were posed, is not on topic for any site on the Stack Exchange network. We don't have sites for all possible topics, at least not at the present time (you can propose new sites via the Area 51 site). Moreover, certain rules as set out in the FAQ are hard and fast rules that apply everywhere on the network. Questions that run afoul of these rules don't fit anywhere just by their nature. Examples of this include questions seeking a recommendation, such as a for a book, website, programming language, or brand of computer. For more on the types of questions you should not ask on Stack Overflow (or other sites in the Stack Exchange network), please browse the answers to this question.