That's a good question. I can only guide you in the right direction here, as even I don't know the correct way to go about it. It sounds like you might want to do something like PXE booting (booting over a network). One caveat that I am aware of: it has to be a wired connection. Beyond that, I haven't really done much research on it as I've never had the need to do such a thing.
You have to set up the host computer really specifically as well. I believe it has to have a TFTP server installed to work at a bare minimum.
I do know that you can use PXE boot to boot the Ubuntu installer, so I don't see why you couldn't use it to boot the whole OS. However, the installer runs totally in RAM, so you'd probably have to make some heavy modifications to Ubuntu to get it to run the full-fledged OS from a remote disk. Frankly, I don't even know that it's possible, but one thing I've learned in all my Linux years is that if you give it enough effort, anything's possible ;)
I was going to suggest a VNC server, but more than likely you're going to be running Windows on the main machine and not Ubuntu, so that wouldn't work out very well. However, you could install it in a virtual machine and set it as if it's just another computer on the physical network. Then you can, say, boot an Ubuntu LiveCD/USB from the dead computer and connect to the virtual machine hosting a VNC server, effectively making your HDD-less computer a thin client.