Well, that's an effect of increasing bandwidth for the internet (what version are we up to on that, by the way?) combined with the advent of more advanced memory devices which are making the need for rewritable storage devices less pressing for most users.For instance, a lot (a lot, a lot, like hundreds of gigs by now) of the data I use regularly is in read-only media. Whereas the things that I regularly edit tend to be only a couple of megs if that (I mean, if you added them all together they certainly wouldn't be even one Gig). On the other hand, a lot of the software I use is either downloaded off the internet in the first place or is regularly updated (or at least can be regularly updated over the internet).
Since the speed of internet downloading and frequency with which much registers software should/must be updated are both increasing, eventually it won't be necessary to keep more than a small amount of the user data for the program on your personal machine. So there really are several factors combining to make it possible to have useful computers that do not have local storage, only memory, processing, and a connection to a network. We see this already in many of the mobile phones and network capable PDAs which have more capability than computers of just a decade ago.
On the other hand, desktop consoles will probably always have disk drives, for several reasons. For one, disk drives are getting cheaper relative to the amount of storage you "need" even though the storage you "need" is increasing by leaps and bounds. For another, it's possible to have a level of data security that just can't be achieved with an internet resource. Then again, more people are using their desktops as fileservers or development stations running/creating/storing at least some unique programs/content all the time.
So we'll continue to see ever more powerful devices that don't have local storage. And a lot of people won't need a device that has a hard drive (or anything similar). On the other hand, as storage becomes cheaper and smaller, it will be easier to include it in devices that don't really need it, and you'll always have users who do need their own storage.
In the end, memory and storage may converge on a solution that is faster than current memory models and yet cheaper and more compact than hard drives (I'm thinking of some kind of nano-scale molecular information bearing device, now). When that happens, we could say that the hard drive is dead, but it would be just as accurate to say that memory is dead. Or even that the processor is dead, since it certainly seems possible that with such a material you'd radically distribute the processing cabability throughout the storage/memory medium itself. You know, the way it is in a brain