It is possible. Anything is possible. But I think you would have to be quite exceptional as teaching yourself all of Bio and all of Chem and preparing for the MCAT without a class structure would, I think, take an enormous amount of self-discipline.
I tend to find that the structure of a class not only helps my discipline but also helps direct my learning. It is the fear of exams that makes me really learn things, and the experience gained sitting those exams (which are never quite the same as questions in the back of the book) which helps me understand whether I *really* know the material or not.
You may be able to do this but I know that for me such an approach would be a disaster, especially when I had no one to help me when I did not understand a topic. Sitting listening to a lecture is much less "effort" than plowing through a book on your own day after day....
Good luck either way!
