Ok, heres the story
Lets say you build a tunnel straight to the center of the Earth, build a 1 cube mile of air (using cooling and solidifying techniques on the molten, of course) in the center. Now theoretically you would just simply float, since the same ammount of gravity is pulling on you from every angle (excluding the pear shaped earth [or banana shaped earth]) So there would be a kind of no gravity around you (exept when you float to one side of the chamber, you will start floating towards the middle since there is more gravity on one side than the one you are on)
Now here comes the question
Ive heard that there would be just wheightless ness. You would feel no pressure. Why is that? Its not like your in 0 gravity, just that gravity is pulling on you from every angle.
If you were to dig a tunnel straight to the center of the sun, you would simply explode from the incredibly gravity pulling on you, correct?
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Lets call ourselfs the Pirate Monkeys!... Its an awsome name...
About the earth question, the earth gravity is constantly pulling on we, so in the center of the earth you would just feel ingravitation.
About the Sun I'm not sure, you can try to find info in google about that.