Note that the thermal time constant of a given geometry scales as the square of the radius. This is why subsurface geothermal heat is there even though it was "charged" gradually over millions of years. So it's not hard to achieve very long time constants just by making the pile a bit larger. They've done these computations. The result is the minimum practical size is a pile that provides ~100 kW(th) constant output.

So the idea is not practical for storage for an individual home, but could work on the scale of a village or small industrial enterprise. It would allow a solar microgrid to be completely disconnected from the larger grid, even at high latitude (assuming one could get a sufficiently good heat engine to work at that scale; it's a bit small for steam turbines.)

If you look at very large piles of dirt, like large earthen dams, if arranged in a suitable geometry they would have thermal time constants (assuming no fluid flow through them) in the thousands of years.

A computation for you: compute the cost of a given R value per m^2 for various materials. Dirt is very cheap, so even if it's not the best insulator it comes out on top. You need the thing being insulated to be large enough that you can fit the thick dirt layer around it, of course.