We used to have exactly this in the UK, back in the day. They were called Storage Heaters and we had a special "Economy 7" tariff where energy was a lot cheaper at night when demand was the lowest.

So basically floor heating :). Don't forget to use some heat pumps to tripple your energy gains

Why heat concrete blocks? Doesn't water have a higher thermal capacity?

Concrete blocks do not leak or wear out. They are cheap and zero maintenance. For more thermal capacity, use a bigger block. You can even make them structural - using concrete for walls.

Bake yourself, your pets, and your belongings during the day, and freeze at night, so that the average temperature, which you will experience roughly twice in a day-night cycle, is comfortable?

Depends on the actual concrete. It will have lower thermal capacity per weight, but that may be offset with higher density. (The most common concrete would be worse than water with capacity being 1.3 (vs 4.2 in water) and density of 2400 kg/m3).