So, I got interested, and did some calculations.
In my neck of the woods (50 deg North, not as North as Sweden), I get a reasonable amount of sunlight. Let's just imagine that 1/3 of the days during winter I get 5 hours of sunlight. I'm going to ignore electricity and just concentrate on heating, because heat is way easier to store. From my meter readings, in the depths of winter I seem to use an average of 2.2kW-worth of gas, most of which goes to the boiler for hot water and heating. Some of it would go to cooking, but I'm guessing not much.
So, if I wanted to collect that much heat from solar, assuming I get 5 hours every 3 days, I would need to collect 2.2×72/5 = 32kW while the sun is shining and then store it for use later. Sunlight is just over 1kW per square metre, so I'd have to have about 32 square metres of parabolic mirrors concentrating light onto a black water pipe. The South-facing side of my house is easily that big.
Then storing it. Water is extremely good at storing heat. Water stores 4194J for each kilogram for each degree centigrade in temperature. If I were to get five 2.4m×1.2m×10cm polyurethane foam insulation boards, I could build an insulated box capable of taking a 2000l hot water tank, which could be put in a shed or something. If that were heated to 90 degrees centigrade, it would be able to return 80kWh of heat before dropping to 55 degrees (which is the lowest acceptable temperature to run a heating system at, before you get Legionnaire's disease). The insulation would lose about 250W, which is small compared to the heat demand from the house.
Yes, there would be times when the stored heat runs out. A sensibly-designed system should hopefully be able to fall back onto the existing gas-fired heating when that happens.
So, these numbers are all very feasible. It'd cost a fair bit to achieve, but it would probably pay itself back within a couple of years, which is way quicker than PV. One of the key factors in that is efficiency and cost. Collecting solar heat is very efficient and uses cheap materials. PV is expensive and still not very efficient.
There are three main requirements for such a project: Enough sunlight when it is cold (and you can adjust the collecting surface area to compensate, so this basically means anywhere outside the polar circles). Enough space to put all the stuff (and as a corollary enough patience from family members to put up with it). And finally a suitable control system that pipes the heat around in an intelligent way, prevents the water tank being over-heated, falls back on gas when necessary, and satisfies the various building standards laws.