No, that isn't how it works. A letter of recommendation should be:
tailored to the recipient, or at least to the type of recipient (grad school, job, internship, etc)
written (or updated) by the professor at the time it is sent
sent directly from the professor to the intended recipient, without passing through your hands
Your suggestion would fail all three of these.
1 is pretty self-explanatory: internships, jobs, grad schools, scholarships, etc, are all looking for different things. You want the professor to be able to focus on what's relevant. A grad school might need to know about your skills in long-term theoretical research; an industry employer might only care about your ability to turn out satisfactory practical work on short deadlines. A letter that addresses the wrong aspects will be useless or worse. The professor doesn't want irrelevant letters going out under his name, either; he'll look like an idiot.
For 2, every recipient of a letter wants to know what you have done lately; usually (hopefully) that will be your best work. You don't want an outdated letter that misses your latest and greatest accomplishments, and a letter that's several months old won't be taken very seriously by anyone. Moreover, if the professor gives you a "generic" or undated letter, it's a blank check: even if you flunk all your classes, or get thrown out for cheating, you'll still have a letter with his name saying "SquarerootSquirrel is doing great!"
For 3, these days, most grad schools / employers / etc will want a letter sent directly from the professor, usually submitted via an online form. Everyone will feel more comfortable that they're getting the professor's honest opinion if you never have any opportunity to read (or alter) it. Based on this alone, I think most professors would simply refuse to put their letter in your hands, even if it were sealed.
But what you can do is to talk to the professor, tell him about your long-term plans for jobs or grad school or whatever, and say that you may be asking him for a letter of recommendation at some point. Ask him if that's something he'd consider, and if there's anything in particular you need to know or do (he may want a copy of your resume / transcript / etc). Then he can start paying closer attention to your work and thinking about what to write, or even start drafting a letter.
When the time comes for you to actually apply somewhere, tell him about it (at least several weeks before the deadline). Then he can just update the letter with your latest achievements, edit to focus on what's relevant for this recipient, and send it off. That part is very little trouble, even if you are applying to many different places.