Ye, it is possible, and I'm going to tell you why and how. So hold on. First here's why you don't need a hard disk: All you really need to run a computer is an operating system, which can access and use the computer's processor and dynamic memory (RAM). It really doesn't matter where this operating comes from. The Hard drive is just a place to store the operating system, software and other information when it's not being used. In theory you can use any storage media to store all this. Guess what, you can do it in practice too. It's just a simple case of writing software designed to run off your desired storage medium and not the hard disk. No, you don't have to do it yourself. People have done it already.
My latest discovery is a beautiful piece of technical wizardry that goes by the unassuming name of Puppy Linux. it's a Live CD with a difference. Live CD's allow you to run an operating system straight off the CD. Unfortunately most Live CD's take the place of the Hard Drive a bit too closely, i.e. the software remains on the CD and is loaded into the RAM as and when needed. Puppy Linux turns this on it's head. When the CD boots, the whole OS is loaded into RAM. This is possible because the Operating System along with a variety of graphic tools, office software, multimedia programs and a full internet suite has been squeezed into a mere 60MB. So if you have a computer with just 64MB of RAM, you can run Puppy with ease. And the fun just starts there. Once the computer has booted up, you can actually take out the Puppy CD and put in another CD, without so much as missed byte. The Operating System simply doesn't care because it's all free of storage media. So you can listen to a music CD or even watch a DVD. And when you're done working, just pop in the Puppy Linux CD, and before shutdown any new files that you have created and any modfications you have made (including new programs installed) will be written to the free space on the disk. Now you not only have your documents on a portable CD, but your whole Operating System. (Yes, if you have a Hard Drive you can save files to it). And If all this CD swapping isn't your thing, your Puppy can even hitch a ride on a USB stick.
So what you are waiting for? Trash that hard disk and go adopt a Puppy today!