You have serious reading comprehension problems. I said it would cost a lot, twice. Twice I said it would cost a lot. See? I didn't say it would cost double.
And you're a bit confused about how electricity works too. No, not "at best the week before." It depends entirely on how over-provisioned the battery is and how over-provisioned the solar array is. In my area, the average is 4.8 full sun hours per day. A cloudy day cuts that to 15%, so the equivalent of 0.72 full sun hours. If I over-provision the solar array by a factor of 7, I don't have to use any battery power at all, even on a cloudy day. Of course that's a bit silly and takes a lot of space. Similarly, I could have stored the power to use on a cloudy day a month ago, if the battery bank is over-provisioned to the point of being capable of storing a full month of power and the solar array is over-provisioned just enough to provide my daily power plus some charge for the battery. That would actually physically fit in my basement. In fact, it would fit with room to spare. For my worst case month (which is in the summer, not the winter), it's only 20 fridges worth of batteries. That fits with room to spare. On the other hand, the factor of 7 over-provisioning of panels would not fit on my property.
But over-provisioning either one by such a huge amount is unnecessary. As I said, the average for my area is 4.8 full sun hours per day. The base system is provisioned with that in mind, and then over-provisioned by any arbitrary amount I care to hedge. If I fail to provision enough for a worst case number of consecutive cloudy days, then yes, I run out of power and the heat stops. In my region, that's only uncomfortable. Only the coldest of cold snaps make my house so cold that it even endangers the pipes. It certainly doesn't endanger life. I've gone two full winters with no heat but the waste heat from my computer, so a few hundred watts. Uncomfortable, but proven possible. So if I'm foolishly cheap, I can provision no storage at all and seriously under-provision the panels, and still my house is habitable. I don't propose to sacrifice one iota of comfort or convenience, and it is still possible.
Expensive, but possible.