I'm just going to add a little since I'm worried you're misunderstanding DaveC.
You can think of gravity as being like an effect. Anything that has mass (anything!) has it's own 'value' of gravity - the value of 'g' is dependent on the object. The bigger the mass of the object is, the bigger the value of 'g' - and since gravity is attractive, the more other objects will be attracted to it. Like Vorbis says, if you think about a toothpick floating around in space - it will have it's own value for 'g', just a very very small one. We don't notice things like this on Earth because the value 'g' Earth has is many, many times larger.
Thus since your body has mass, it too has its own gravitational field - the only reason you don't notice is because you're so small in comparison to the Earth. Gravity is prevasive and for these purposes can be considered to 'go through' anything. Consider yourself falling through the sky. Now, before you activate your parachute(!), you already know that you'll be experiencing an acceleration of 9.8ms/2 - if gravity didn't affect all of your body at the same time, what would happen?
If the outside of your body shielded the inside from gravitational effects then it would mean that your skin would be accelerating at 9.8ms/2 whilst, rather unfortunately, your internal organs would be at 0ms/2 - thinking about this will reveal the fact that it doesn't make sense. Where would your insides go? From common experience, they need to move at the same rate as the rest of your body.