I second this - a few years back I switched my home server to a Mac Mini from an old x86 box, for power draw and space/noise reasons, much like the original poster. At the time I checked out alternatives, but there wasn't much to recommend other machines, all the ones I could find had much more limited storage space. No doubt that has improved recently, but being able to fit any standard 2.5" drive is a big advantage if you want to use it as a file server.
My Mini draws 20W when idling (I tested it with a kill-a-watt). Power use will be higher under heavy load, of course, but your average home server spends most of the time idling. I'm pretty sure the 85W/110W ratings are the maximum the PSU can handle, not the power draw you'd expect in normal use. My box runs a web server, ssh, mail server, file server and various other bits and pieces. X is not installed. It is one of the old PowerPC Minis, which I think draw a bit less than the more recent Intel Minis, but I can't imagine the power draw has increased that much.
My advice to the OP would be to pick up a second hand Mini and use that - there might be machines out there designed specifically as low power home servers, but Minis are fairly easy to come by and easy to install Linux on as people have been doing it for a few years now, even if Apple don't encourage it. If you're thinking about environmental impact as well as your electricity bill, buying a second hand machine is going to be better than buying a new piece of kit. This was another part of my decision to go with a Mini, there are various computers designed to do the sort of thing the OP has asked for, but they're much more niche and consequently hard to find second hand.
By the way, if you choose to use a PowerPC Mini, choose a distro that fully supports PowerPC! When I set up the box Ubuntu still officially supported PowerPC, but it has since been switched to unofficial ports only support, which is pretty flaky. Debian is a much better bet, I am now using that as it is much more reliable (note to anyone who wants to call me on this, I am very happy using Ubuntu on x86 desktop, but my recent experiences of the PowerPC releases have not been favourable).
Some people are suggesting laptops, but I wouldn't recommend one myself. For one thing, they aren't designed or expected to be on all the time, and I suspect you're more likely to run into heating and dust related issues. For another, one of the main advantages of a laptop is that it has a battery and therefore won't require a UPS. However, leaving the machine constantly on and charging is going to kill the battery life fairly quickly, at which point it's not really very useful. On top of that, most laptops use 80W+ when running on mains power. They're usually only designed to save power when running from battery. Obviously you can change the power saving settings, but it's going to be a pain to do so.