Is it possible to store a large enough amount of power to supply a house?
It is probably the cost, rather than the size, of the battery which will limit how large a battery you'd be likely to install in practice.   This site claims the "true cost" of a battery only a third the capacity of what Dave Tweed calculates would cost around $58k for Li-ion or $105k for lead-acid (converted from Euros to dollars).  They claim lead-acid would actually be more expensive due to having higher transportation cost, shallower usable depth-of-discharge (DOD), and much shorter cycle life which would lead to multiple replacements over the same time period.

http://www.powertechsystems.eu/en/technics/lithium-ion-vs-lead-acid-cost-analysis/

In practice you might not install enough energy storage to run for "a week" unless you had good reason to believe your offline generator e.g. "windmill" might produce little to no energy over that long a time period.  The cost could be substantially reduced by only storing 1-2 days worth, or even less depending on reliability of the source of generation.

For example, with a solar setup, the house might be powered "directly" (not counting inverters etc) from the solar panels during the day, with enough extra solar capacity to also charge the battery during the day; the house would only run from battery at night.  Thus the battery capacity required would be enough to get through 1-2 nights and maybe a few especially cloudy days.  As energy usage tends to go way down while people are sleeping, capacity required to run from battery at night would be much less than day.  Paying particular care to reduce usage, e.g. efficient appliances, LED lighting and so on would further reduce cost.

The assumption here is that generation is cheaper than storage, notably for solar, which has gotten much cheaper the last few years.  If you needed more energy to run from during the day, you'd install more solar panels rather than more batteries.  The batteries are just for when the solar panels aren't getting enough sun to power the house.