I meant ‘bare minimum’ in the sense of what's needed for the system to be profitable, not in the sense of ignoring costs.

The problem with using this approach for daily cycled loads is that it relies on passive heat transfer to distribute heat through substantial regions of dirt. This simply doesn't work for daily storage.

You can overbuild, but then your energy losses are going to be immense, because you never saturate or drain the bulk of the material, and are just losing energy to it.

You could build faster cycling systems instead, and active systems especially can cycle reasonably fast, but then your dominant costs no longer reduce down to a pile of dirt with a few rods stuck into it.

You would need more rods if you're cycling more of it at once... but you can add 25x as many rods and it'll still be 20x cheaper than a battery, right?

I don't think losses would be immense. You'll spend a few months warming up the neighboring dirt, but after that the amount of heat escaping per day will look about the same as the seasonal system.