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I understand the previous responses to other answering if you were at the center of a planet, would you be weightless? Yes you would due to the forces of gravity pulling at you from all sides.

However this is different question in that if you proceed down a shaft moving towards the center of a planet there is a location somewhere down the shaft (well before arriving at the center) where you have as much mass above your head as below. At that point would you not have gravitational forces have equal gravitational pull above, from the sides and below resulting in your being suspended between the equal forces?

BH Carney
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There is no gravitational force on an object which is within the confines of a massive spherical shell, no matter the thickness of the shell layer. Newton proved this.

This is anywhere within the shell, not just at the center. In your case, since there is only the shaft, the sphere is a solid, so gravity would only be completely negated at the center of the planet.

Jiminion
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You can solve this through an application of Newton's law of universal gravitation, or equivalently Gauss's law for gravitation, but one simple way to see the answer to this is through an argument by symmetry.

Let's say that you are truly at the center of a solid, spherically symmetric sphere of mass, and the gravitational force is nonzero. Which direction would it pull you?

The only way for gravity to pull on you without picking out one special direction in space, is for the force of gravity to be zero. And so it is.

Brionius
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There is a gravitational force anywhere in the universe but it happens to be balanced on all sides for the situation you describe so there would be no acceleration due to gravity. The evidence of the gravity is the extreme pressure, density and heat.

Bill Alsept
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