If you had a volume of 1 cm³ of each material:
Water will have a mass of 1 gram and can store about 4.18 J of heat per degree Celsius.
Iron will have a mass of 7.87 grams and can store about 3.54 J of heat per degree Celsius.
Sand will have a mass of 2.2 grams (about) and can store about 0.703 J of heat per degree Celsius.
The proper billionaire bunker would have an underground silo filled with iron balls about six inches in diameter. A solar collector would heat air to 50 degrees C or more, and blow it into the bottom of the silo. Heat is transferred to the iron balls as air passes through. The silo has a diameter on the order of three meters and is three meters tall. Imagine a giant jar of Jelly Beans. Residential heating pulls air through the the silo as needed for heating. A second silo could have night air blown through it to provide air conditioning in the daytime.
I was thinking of building a thermal ballast under the house consisting of six inch cobble stones in a wood box about ten feet square. (Years Ago) A solar collector would charge it with hot air at the bottom, and air for house heat would be drawn out of the top of the wood box. All that is needed is solar power and enough electricity to blow low speed fans to provide air flow. Solar power to operate the thermal charging air flow is an obvious choice.
But what would have been the point? Mrs. Dog would have seen it as a total waste of time and money. A ten foot square box of rocks that could cause a divorce, and then I would not be in the house.
Cultural hegemony kills many a dream.
When someone is high on hopium they like to say there are all kinds of solutions 'if people only had the will to carry them out'. Usually when they say this BS I think their fingers must stink from the shit they are pulling from their ass. But solving building heating issues by thermal storage is one of the imagined fantastical solutions that actually exists.