You need a lot more than a cubic kilometer per satellite, because you need an entire orbit around the planet. You can't have one row of satellites orbit at the

You need a lot more than a cubic kilometer per satellite, because you need an entire orbit around the planet. You can't have one row of satellites orbit at the equator, and another row orbit at 40 degrees north latitude. A row is already difficult, they catch up with each other quickly just due to random perturbations and elevation differences. Instead, each wrap of your ball of yarn crosses over the others. Each row intersects most other rows.