I have a Mitsubishi Hyper Heat, and it operates to -13f, which would be unheard of in Western Oregon (where I live). So long as your house is well insulated, you'd likely be fine with a heat pump. Pay attention to R-Value for insulation, but if you choose the right materials, that goes a long way for heat storage.

I know there exceptions in Minnesota where it can get considerably colder than -13f, but I don't know how long those super low temps are sustained. You would definitely need some sort of emergency heating system. An outdoor propane tank (the larger variety) and a propane heater would likely do the trick as an emergency backup. Not great for air quality in your house, but plenty of people use that as their normal heating.

I want this someday too (also live in MN). Wood pellets or just a wood burner are your best bets for secondary heat. I grew up in WI and we only had electric base boards and a wood burner (then pellet stove). Get the house up to 85 before bed and by morning it's in the high 50's and start the fire again. You need to be ok with fluctuating temps to really make it work.

You mention cloudy days but the one positive about our brutally cold winters is that when it's below zero it tends to be sunny out. If you set up your solar to extract as much sun as possible during these times, as well as build your house with large south facing windows with stone or concrete flooring you will not need to heat the house much during even the coldest days (solar radiation will heat the house and the concrete flooring will release what heat it gained in the evening).

Keep in mind that you don't need a traditional electrical battery for heat - storing heat directly is extremely cheap, you just need insulation and thermal mass (i.e. sand/rocks, any old crap will do). There are a fair few systems that use easy summer-solar to generate heat, store it for months at a time and then release the heat in winter.

It scales up really well thanks to the cube-square law, which is a euphemistic way of saying it's hard to make a viable system that's really small. But if you're off-grid because you're in the middle of nowhere, then you can spare a few square meters anyway.

Look up ground-sourced heat pumps. You store your heat in soil during the summer, and draw it out in the winter.

If you do some back-of-the-envelope math on this, you'll notice that this will be cost prohibitive to implement using batteries.

What about a wood burning stove? That would be more reliable if you are truly off the grid.