Cut the Rope

Imagine a rope that runs completely around the Earth’s equator flat against the ground. You cut the rope and tie in another piece of rope that is 710 inches long. That increases the total length of the rope by a bit more than the length of a bus.

Now imagine that the rope is lifted at all points simultaneously, so that it floats above the Earth at the same height all along its length. What is the largest thing that could fit underneath the rope?