Frankly quite small. I'm running much more than a basic FreeNAS on a p4 1.5ghz, and it does quite well. I'm obviously limited by the 440BX PCI bus, but it serves standard def video to my windows pc quite well. (no HD for me yet, I'm still in the stone ages apparently)
For HD, you probably need to get into a dual core atom or better. These will have a pci-e bus, faster ram and sata 2 for drive access. I'd say a minimum of 1gig ram for a barebones setup.