Disclaimer: Am retard with only BBT-tier knowledge of science

So, if large masses are supposed to bend space - resulting in gravity - do they also "stretch" it? When the effect is visualized in CG animations, or with balls on stretched lycra, the space-stand in is clearly deformed, making it shaped like a cone, rather than a circle, and the surface of a cone is greater than area of a circle.
Does it mean that there is "more space" near large masses than away from them? If so, does this mean that the volume of superheavy objects in space has to be calculated in a different way, as it is greater than just a volume of a sphere? Are they "bigger on the inside"?