Hello.

I'd like to try hosting a low-traffic / no-traffic website from my home. I'm not concerned at all about CPU horsepower or accelerated graphics. This is just going to be a box that runs apache and sshd.

My main concerns are initial cost, power efficiency and ease of installation.

Ideally I'd like to be able to install Ubuntu, but Debian or Fedora are also acceptable.

I'm guessing that a laptop would be the most power efficient (with the display turned off), but maybe I'm wrong. Is it possible to build a diskless system which uses a flash drive for primary storage?

What kind of low-power hardware is out there (and is affordable) and is supported by the major distros?

Thanks.

A laptop is not designed to run 24/7.

Seems to me that a sub-GHz P3 would work nicely. Mine has Intel MOBO with on-board graphics. I added an AGP accelerator, but you would not even need that. I've had numerous Linuces on that box with no problems.
Used P3 boxes should be pretty cheap......

Well, after reading your first sentence, laptop came to mind and apparently you've considered it also. But generally laptops don't displace heat as well as a desktop, you would/should pull the battery out and use AC current only to save your battery from the constant heat. In that sort of scenario, you would probably go through 3 or 4 lappy's compared to a desktop's life expectancy, especially if you live in a warmer part of the world without air conditioning. So you would have to weigh the difference, the cost of 3 or 4 lappy's compared to the extra power consumption cost associated with a desktop, which can be tweaked to cut the cost also.
I'm not sure about 64 bit processors, but AMD processors usually cost less to run and run a little cooler. That was the general consensus a year or so ago, I'm not exactly current on today's generation of processors.

I had a laptop that ran more or less 24/7 for three years only with an occasional reboot and holiday. The longest period I went without rebooting was about three months and the only problem I encountered was a crashed hdd in the end.

I always thought it caused more wear and tear to turn on and off the machine all the time?

Most machines built in the last 2 years or so run very efficiently. Obviously the fewer components you have installed the more cost efficient it will be. So leave out the video card and whatever else you don't need.

If you want to install something like Ubuntu, you'll probably need at least a 500Mhz system with 3Gig harddrive. Otherwise, just buy the oldest crappiest computer tower you can find on ebay, it should be fine.