Very fun thought experiment. It reminded me that we did a problem in calculus to compute the effect of gravity inside a hollow sphere. It turns out to be ZERO.

So, if the Earth's mass, was all densely concentrated in a, say, 1 mile thick shell, you could drill a hole through the shell and experience total weightlessness when you popped out on the "inside" (assuming a total vacuum on the inside - if not - you'd experience a very small gravity toward the center based on the mass of the contained atmosphere).