In my WIP, I have a thick, industrial extension cord going off a third floor balcony, and down into a basement, then into a passage several hundred yards long.
Is their a limit to over how long a distance you can chain extension cords together?
Batteries won't work, they need a lot of power to run something like a compressor, but I don't want to put in something that is impossible. Implausible I don't mind.
It's not impossible, but could be implausible. Variables are the length of the cord (longer is worse), the size of the elecrical conductors in the cord (larger is better), and the number of cords (each plug/mechanical connection is a source of resistance, more is bad). You can (or you could years ago) get 100-foot extension cords at Home Depot, good for things such as electic lawn mowers (I think someone still makes them). A compressor could probably run on a few strung together, but the more there are, the greater the voltage drop in the cord, and the less likely the compressor will start up (it uses much more current when it starts up than when it's running). Also, if it does run, running it on reduced voltage could cause it to overheat (electric motors are funny about that - LESS voltage can make them use MORE power, and turn it into heat).
Here's a scenario: they have two things plugged into the end of the cord: a 100 watt lightbulb and the compressor. The light is on, and the compressor is turned on. The light goes to about half brightness, and the compressor hums and groans but doesn't start running. Someone who knows something about electricity sees what's happening, and knows the light is pulling power too. If it's turned off, there's a little more power for the compressor, and it's more likely to start up. They turn the light off, start the compressor, and then turn the light back on. The light slows down the compressor a little bit, but since it's past its super-high-current starting phase, it keeps running.
Perhaps something more technically plausible is they buy a 500-foot roll (in your WIP it may take several rolls) of Romex (brand name of house wiring - stiff wire meant for permanent in-wall installations rather than "portable" use) - this roll will weigh maybe 100 pounds and at current copper prices probably cost well over $200 (so perhaps they steal it). They fix plugs and jacks on the ends and do their thing. This will work better than extension cords, but even that many extension cords will be expensive and heavy. Check your local hardware/home improvement store...