Possible, however very difficult. First off, if you want it to be constantly on, you're going to definitely sacrifice performance. Your clock frequency affects your power consumption hugely, so you're going to have to use something way slower, alix or fitpc2 for example. If you are ok with that, then the next thing would be removing as much add ons as possible: ext HD, bluetooth, and anything external that draws current. Finally get a deep cycle battery. All that together would get you a PC that might run for a couple days without car running.
An alternative would be using WOL wifi cards which only turns on your computer remotely as required.