Ya know, there's a cave over here in the states somewhere, I can't remember what the name of it is, or where it's at, but in that cave, you're near-weightless. There's a deposit of some kind of ultra-dense material directly above the cave (I'm not sure if the geologists ever figurd out what it is or not, been a while since I read about it, I'll have to dig the article up again), which in that small area counter-balances the gravitational pull of the Earth's core.
How do you explain that one?