What are the technical challenges to store electricity from lightning strikes?
I'd argue the challenge isn't technical, it's economical. A normal lightning strike transfers about 500 megajoules of energy (says Wikipedia). Wikipedia also says a single wind turbine comes as big as 7.5 megawatts these days. Thus, a single bolt of lightning is equivalent to running a big wind turbine for:

$$ \frac{500MJ}{7.5MW} \approx 67 \text{ seconds} $$

That's also assuming that all the energy from the lightning is electrical and that you can get at all of it. You can't: most of that 500 MJ of energy is spent heating the air and Earth. If you want to get all of that energy, you will need to make the lighting do work on something that isn't the air, which means very tall towers. All told, there just isn't that much energy in lightning.

If the economic problem weren't there, your approach is basically sound. We already engineer skyscrapers to withstand lightning strikes without damage. The problem is in storing the energy. I'm sure something could be conceived with superconductors, or really big supercapacitors.

Trouble is it's cheaper to put up a wind turbine, or better, dig some rocks out of the ground and burn them.