WIKIPEDIA (VIA EPIA) : VIA EPIA (VIA Embedded Platform Innovative Architecture) is a series of mini-ITX, nano-ITX and pico-ITX motherboards with integrated VIA processors. They feature a small size and low power consumption, so are particularly popular in the embedded market.

Elite C7VCM Mini-ITX Motherboard with VIA C7 1.5GHz Processor - $70.00
12V Power Input Ready

 

 ALIX.1C Board with 1 LAN and 1 miniPCI - 5W - 256Ram/DCtoDC included ..500MHz fanless .. $140.
add $30 for .5w wireless expantion card.
add $10 for power suppply.

ALIX.2C0 with 2 LAN and 2 miniPCI, LX700, 128Mb -- Only Networking (no video/audo/usb/etc) - $120
Also 1 Lan and miniPCI available. (cool encloseures "wrap box" with antena $50) 5watts.

NiMH batteries like the AA kind... are 2.3Ah at 1.2V @ about $1.50/battery
D size NiMH are 9Ah at 1.2V 

So for a 5-10Watt average setup.. with a 10 AA array (12V) of power (23Ah) can run about 2-4 hours.
(note if can run 30-45 mins on just a couple with 5w consuption.)