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It's plausible. The moon's width is 2,158.8 miles, and I could imagine an explorer being >3000 miles from another human

The actual travel distance would be further because you can't just bore through the moon.

I'm too lazy to do the math, but what's the point-to-point distance (through the planet) of points that are 5000km along the surface?

Assuming a spherical earth, it's 4873 km.

Assuming a spherical earth in a vacuum... One of the few times that simplifying assumption matches reality.

Not quite; the Earth is an oblate spheroid. Assuming it's a sphere is certainly a close enough approximation for this exercise, though.

How far is it if you assume a flat earth?

5000km