https://iopscience.iop.org/article/10.1088/1748-9326/ab91e9
In your once in a lifetime event I'd be happy if we used a bit of gas. This is something, after all, that Japan also had to do for years after Fukushima went kaput.
Bear in mind also that demand for heating and transportation can be timeshifted. An Octopus tariff in the UK already allows for this. We've barely scratched the surface of whats possible here because electric grids are slow to evolve to this new reality.
Gas alone does not produce electricity. Gas power stations do. This means building enough gas power stations to cover power requirements of the entire country, then having them sitting most of the time idle there, "just in case".
This is the core of the problem: wind and solar need to be massively backed up. You cannot rely on them supplying even 5 per cent of their nominal capacity if the weather turns against you.
Such a massive backup is eye-wateringly expensive, too. And it consumes concrete and steel, two elements whose production is carbon-intensive.
If you assume:
* We dont already have the infrastructure.
* Power cant be imported.
* Power usage cant be timeshifted.
* Power cant be stored in batteries.
* 10 day periods without wind and sun are routine.
Then yes, it might get expensive.
How can you even estimate the cost of batteries? Insane amounts of batteries would be needed, capable of storing all of Germany's energy needs (including transportation and residential heating) for 10 days or so. That would be at least 10 TWh, as Germany currently uses about 550 TWh per year.
The global production of lithium ion batteries in 2018 was 290 GWh (1000 GWh = 1 TWh), expected to increase to 2 TWh per year by 2028 (https://www.statista.com/statistics/1247625/global-productio...).
In other words, even if Germany would buy every single battery produced on Earth over the course of 5 years, it probably wouldn't be sufficient to provide energy for 10 winter days without wind.