What happens when, teoretically, somebody is in the gravity center of the earth?

It's easy to show (mathematically) that at the center of gravity
(center of mass) of an object, the gravitational force between
that object and any other one is zero.  In other words, at the
object's center of gravity, it's as if that object has no mass.

At the Earth's center of gravity (wherever that is), you would be
"weightless" as far as the Earth is concerned.  (You'd still be
gravitationally attracted to the Moon, the sun, etc.)