I'd suggest keeping almost everything on the computer alone. Obviously you'll want to keep original documents, but you could scan into your computer photocopies of certificates, photos, etc. You could then either chuck out all the non-original paper, or store it in boxes up in the loft for emergencies, if you don't trust the computer entirely.
There are many advantages of having it all on the computer - you can share it more easily, you can have it all on a laptop when you're on field trips, and you can have multiple indexes - alphabetically, chronologically, tree structure etc.
Of course, you then really have to think seriously about backups. Make frequent copies, and don't store them all in your house. Getting relatives or friends to keep copies would be ideal - offline (e.g. DVD) and on their computers (if it's only on computers it could be vulnerable in the case of a widespread virus attack).
Another thing to think about is how to ensure it gets passed on to future generations. If everything is on paper and takes up a lot of space, there's the risk of it being chucked out by some disinterested relative after we're dead and gone. And since copying thousands of sheets of paper is impractical, you're really putting all your eggs in one basket, hoping that the family member (and their descendants) who inherits the papers will look after them. With computerised copies, you can give all the family a copy.
It would be great if some government body (say, the National Archives) could promise to preserve such stuff, perhaps for a one-off payment. You could then specify that it not be open for public access for say 50 years after your death.
There was a discussion about organisation and methods for storing genealogical data a year or so ago:
http://www.rootschat.com/forum/index.php/topic,22138.msg82605.html#msg82605
