You lost me when you switched from work measurable by Joules or Ergs to a different thing measured in dollars.  

Imagine we have $1B of energy we want to store for 100 years. In other words, we can spend $1B worth of resources today to create $1B worth of value, and store it in whatever form that can retain all or most of it for 100 years. What are our best options?

Buy an oilfield. Oil has stayed in the ground for millions of years, it'll be fine for another century. And you'll get more energy from it in a century than you can get today for the same money, because extraction technology will have improved by then.