Area of gravity at the center of large, dense celestial bodies...

I've read that at the center of large celestial bodies there's zero gravity (or close to). While confirmation would be nice, if true, I'm wondering how large that area can actually be and moreover, does it scale up with more mass and/or even size - that is, does the sun have a larger center area of low (zero?) gravity than the earth and so on with evermore mass. Or is that area the same regardless of mass' size?

Thank you