A new version of this page is released at alternative laptop battery power.
So what's left when there's no mains (grid) around to power the laptop / adapter? Basically it's getting access to DC power (Direct Current), the voltage / amps needed at the DC jack of your laptop. For most laptops it's somewhere between 15 - 22 Volts DC at an Amp rating between 1 to 4 Amps. Important is a stable clean DC Voltage supply because a laptop has no power stabilizer on board. There is little margin to play around with. The most common external DC power source is an external battery, usually a normal car / solar battery.
First is running a laptop from a 12 or 24 volt car / truck battery. For this a DC-DC step-up/step-down converter is required. Usually it's a digital "switching" power supply using an internally generated sine wave sliced up into a stable DC voltage. There are many on the market already, with jacks for many laptop brands. Important is to know what you're doing. Getting DC voltage polarity wrong fries the laptop chips instantly and irreversibly.
Secondly is running a laptop from a DC-AC power inverter from a 12/24 volt battery, generating 220 / 110 Volt mains. The conversion does cause a extra loss in power but is usually preferred above the DC-DC converter because of it's easy availability and less technical implications.
Third and less well known is to create a (bigger) battery cell matching the voltage needed by the laptop. Like using a 12 Volt car battery in series with several 6 Volt motor cycle batteries in parallel. (creating 18 volts) This is no-go area for non-electricians and can only be done effectively with careful experimentation. However it eliminates any DC-DC conversion and gets the best electrical efficiency. The batteries can be charged with either cheap starter battery charges or more sophisticated battery chargers from the RV / Marine market or eco / green power. Most important is to NOT charge batteries when the laptop is connected!
Stable DC voltage (without mains) comes from (external) common 12 volt batteries. Most accessible are 12 volt car batteries or deep cycle solar batteries. Roughly a common 12 Volt battery supplies about 90 Ah on average. That's less than 1 KWh from full to empty. A car battery won't last many deep charges, so a deep cycle solar battery would be the only sane option. And usually empty means 30-40% left of the 90Ah which makes the effective capacity about 600 Wh. That's rhoughly 12 hours of continous use - after that your car probably won't start anymore.