I think a lot of participants in this thread are confused—I hope I’m not one of them! The gravitational force on an object placed exactly at the center of a spherical body is zero. If you could somehow enclose yourself in a protective bubble at the Earth’s center then you would float, weightless, just as you would in outer space.
One way to explain this is to consider the Earth to be a nested series of concentric spherical shells. It’s well known that the gravitational force inside a spherical shell of mass is exactly zero. Thus as you descend below the surface you only feel the gravitation of what lies below you (toward the center) but not above you (toward the surface). When you reach the center there is nothing below you so your weight becomes zero.
Indeed if you bored a hole straight through the earth’s center and jumped in, your weight would decrease linearly until you passed through the center, then it would increase linearly again until you emerged on the opposite side.
One point of confusion in this discussion is the notion that at the center you are somehow pulled outwardly in all directions, being stretched apart, even if weightless. This is false. It would be as if there were no gravity at all—no falling, no stretching, no gravitational force at all.
(My earlier mention of black holes was in connection with very large tidal forces that stretch you out—in sharp contrast with the situation at the Earth’s center.)
Another point of confusion has to do with pressure. Gravity is zero at Earth’s center, yet the pressure is at maximum—in fact it’s calculated to be millions of atmospheres (ref). That’s why the inner core is solid iron despite being way hotter than iron’s normal melting point. You’d be instantly crushed into a greasy lump. I wouldn’t describe this merely as “atmospheric pressure,” however, since it arises from the combined weight of atmosphere, crust, mantle, and core.
It might seem counter-intuitive that gravity could be zero while the pressure is enormous, but really one has nothing to do with the other.