The answer depends on how you mean the question. In practicality, with the center of mass at roughly 4,000 miles away, even an airplane five miles up represents a change of only around a tenth of a percent, and other effects could easily overwhelm any difference.
In theory, the answer is a little more complicated. At the center of the earth, you would "weigh" nothing. The gravitational pull would be equal on all sides. You would need an impossibly strong and heat resistant and likely radiation-proof protective bubble though.
What can be a little mind-boggling at first is the fact that you can never COMPLETELY escape Earth's gravity. Similarly, we are slightly affected be the gravitational pull of EVERYTHING in the universe that has mass, no matter how far away or small. That explains those days when you feel like you are pulled in all directions.