Building a LAN on in the woods
There's been much mention of inverters and such for powering the router. From an efficiency standpoint there's likely an even better approach considering the actual power you need.
Most SOHO routers are supplied with DC under 12 volts, so you would gain by getting an adequate converter/regulator to go from standard automotive 12 volts (usually 13.8v) to the appropriate input voltage (likely 9 volts). This would probably save you a lot of juice simply not having the chain of voltage conversion and de/modulation (i.e. 13.8vDC => 120vAC => 9vDC).
What about some 9 volt power tool batteries wired up in parallel? You could probably run a solid state device for quite a while considering those batteries are designed for high draw motorized tools. I remember my dad used to run our first VHS video camera on Makita batteries.
How about a battery powered wireless router:
http://www.cradlepoint.com/products/phs300-personal-wifi-hotspot
Some spare batteries pre-charged and you will be good to go. No wires needed - at all.
Have you investigated solar panels (with a battery)? a wireless router doesn't pull that much juice, so you might be able to tie a solar panel into a 12v battery, and power the router off of that. It would also be silent, which might be a plus, depending on your reasons for building a network in the woods.
A few weeks after writing this answer, I cam across this link to Popular Science, detailing how to do just this.
Freeplay Energy sells a product called the Weza which is a foot powered charger with battery. It has a 12v adapter that you might be able to plug into with a converter.
I'm not sure about Windows but if you use a Mac you can get a 3G connection and then use Network Sharing through WiFi to create an AdHoc network. This is how we do roaming demos sites. We use one Mac as the 3G up-link machine and the others connect to it over WiFi.
If you have line of site you can go quite a long way with standard wifi equipment. Look at what these guys have done as an example.