I am not sure batteries would be a problem for a household robot. There are power sockets everywhere. And the robot can take a break from most tasks to charge itself (and like a dog it would spend most of the day sleeping anyway!).
An advanced household robot would not even have to take charging breaks because it would be well up to the task of plugging itself in wherever it would work (except for the dog walking part, obviously)
In a similar line of thinking, I am actually kind of surprised that there are no wired quadrocopter swarms yet, that would form a bucket line of short cables to trade range for the power and infinite endurance of a grid connection. I am quite sure that there must be applications where this deal would be worthwhile.
To transmit power, its better to use high voltage and low current. Reduces transmission losses; uses much lighter cables. Then where the power is needed its stepped down with transformers or 'buck converters'.
Might still be cheaper (storage, transport, setup, investment) than a crane or scaffolding when you don't need to support heavy loads. And scaling would only be limited by how much power you can push through the chain, whereas sticks need to be quite massive just to support themselves once you go beyond a few meters of length.
It's not a swarm, but I think the one drone from 3D Robotics can be tethered for essentially unlimited flight time. So that's some progress towards what you have in mind.