The area of the Sun and the Earth and every other self-gravitating body in hydrostatic equilibrium is also proportional to the body's mass. Volumes are weird though: unlike area, relativistic volume can depend on the body's (and the spacetime's) history and composition rather than just the body's mass. In general the spatial volumes inside massive bodies in curved spacetime are larger than in Euclidean-Newtonian space.
The volume deviation is carried in the Ricci tensor
The highest-scoring answer at https://physics.stackexchange.com/posts/36411/revisions is a fairly reasonable attempt to calculate the volume deviation for nonspinning ~spherically symmetric bodies with the masses of the Earth (~ 10^2 km^3) and the Sun (~ 10^12 km^3), compared to the Euclidean-Newtonian volumes. Qualitatively, dropping these symmetries and the uniformity of the matter will tend to make the volume deviation larger.
> there is no volume
The volume deviation becomes enormous for compact (relativistic) objects, and for black holes one has to exercise care in even defining a volume, since naive choices of coordinates will show a divergence. Typically the choice of a 3-space inside the horizon has a time-dependency, and most choices of 3-space will tend to grow towards the future.
Christodoulou & Rovelli's (C&R) approach: https://arxiv.org/abs/1411.2854 ("it is large" for the largest volume bounded by a BH's area should win some sort of award for understatement). https://arxiv.org/abs/0801.1734 (reference [5] of the 2014 C&R paper) takes a slightly different path to the same conclusion.
YC Ong (several other references, and a number of related later papers) has a nice article at https://plus.maths.org/content/dont-judge-black-hole-its-are... The prize quote: "To give an idea of how large the interior of a black hole could become, this formula estimates that the volume for Sagittarius A, the supermassive black hole at the centre of our Milky Way Galaxy, can fit a million solar systems, despite its Schwarzschild radius being only about 10 times the Earth-Moon distance. (Sagittarius A is actually a rotating black hole, so its geometry is not really well-described by the Schwarzschild solution, but this subtlety does not change the result by much.)" And: " These examples show that, in addition to the surprising property that the largest spherically symmetric volume of a black hole grows with time, in general, the idea that volume of a black hole grows with the size of its surface area is wrong. In other words, by comparing two black holes from the outside, we cannot, in general, infer that the "smaller" black hole contains a lesser amount of volume. "
The area of a Schwarzschild horizon is straightforward to define, and unique for constant mass. (Procedurally you could count the number of unique tangent planes at r_{schwarzschild}, but there are other ways of arriving at the area).
If your sweater "weave" represents a set of orbits around the black hole and your ant free-fall along those rather than walk, you are getting close to a solution of the geodesic equations for a black hole. A free-falling ant will stick quite firmly to geodesic motion around a black hole. However, there are definitely plunging orbits that will take the orbiting-ant inside the horizon, and there is an innermost stable circular orbit (ISCO) that isn't solid like the yarn: a small perturbation of an orbiting-ant there will knock it into or away from the BH. But an un-knocked ant can circle forever.
The ISCO (3r_{schwarzschild} for a Schwarzschild black hole) is quite a lot of ant-lengths above the horizon of a BH (2r_{schwarzschild}). Spinning black holes have a narrower gap between the ISCO and the point of no return.
The point of no return for a spinning hole is just that: the ant can't backtrack, but will continue moving "forward" from there, and for a massive enough black hole it could do so for an hour or more before it feels the discomfort that precedes spaghettification. The "no drama" conjecture holds that the freely-falling ant won't even notice crossing the point of no return, although astrophysically it is likely to have noticed things falling inwards on different trajectores even above the point of no return (at ISCO around an astrophysical black hole the ant has a good chance of being knocked by something on an intersecting trajectory).
> fabric of spacetime
Misleading terminology. It's not a substance. Spacetime is nothing more than a collection of possible trajectories, and none of them needs to be realized. (Our universe has an enormous number of unrealized trajectories compared to ones on which real bodies move).
> bunched up in a very thin shell
The "thin shell" is just a set of points of no return, and for an astrophysical black hole where exactly each point is can be rather fuzzy since it depends on the outside universe which is filled with moving ants (and galaxies).
The volume deviation is carried in the Ricci tensor
https://en.wikipedia.org/wiki/Ricci_curvature#Direct_geometr...
http://arxiv.org/pdf/gr-qc/0401099v1 (section 5.2)
https://math.ucr.edu/home/baez/gr/outline2.html (bullet point 9)
The highest-scoring answer at https://physics.stackexchange.com/posts/36411/revisions is a fairly reasonable attempt to calculate the volume deviation for nonspinning ~spherically symmetric bodies with the masses of the Earth (~ 10^2 km^3) and the Sun (~ 10^12 km^3), compared to the Euclidean-Newtonian volumes. Qualitatively, dropping these symmetries and the uniformity of the matter will tend to make the volume deviation larger.
> there is no volume
The volume deviation becomes enormous for compact (relativistic) objects, and for black holes one has to exercise care in even defining a volume, since naive choices of coordinates will show a divergence. Typically the choice of a 3-space inside the horizon has a time-dependency, and most choices of 3-space will tend to grow towards the future.
Christodoulou & Rovelli's (C&R) approach: https://arxiv.org/abs/1411.2854 ("it is large" for the largest volume bounded by a BH's area should win some sort of award for understatement). https://arxiv.org/abs/0801.1734 (reference [5] of the 2014 C&R paper) takes a slightly different path to the same conclusion.
YC Ong (several other references, and a number of related later papers) has a nice article at https://plus.maths.org/content/dont-judge-black-hole-its-are... The prize quote: "To give an idea of how large the interior of a black hole could become, this formula estimates that the volume for Sagittarius A, the supermassive black hole at the centre of our Milky Way Galaxy, can fit a million solar systems, despite its Schwarzschild radius being only about 10 times the Earth-Moon distance. (Sagittarius A is actually a rotating black hole, so its geometry is not really well-described by the Schwarzschild solution, but this subtlety does not change the result by much.)" And: " These examples show that, in addition to the surprising property that the largest spherically symmetric volume of a black hole grows with time, in general, the idea that volume of a black hole grows with the size of its surface area is wrong. In other words, by comparing two black holes from the outside, we cannot, in general, infer that the "smaller" black hole contains a lesser amount of volume. "
The area of a Schwarzschild horizon is straightforward to define, and unique for constant mass. (Procedurally you could count the number of unique tangent planes at r_{schwarzschild}, but there are other ways of arriving at the area).
If your sweater "weave" represents a set of orbits around the black hole and your ant free-fall along those rather than walk, you are getting close to a solution of the geodesic equations for a black hole. A free-falling ant will stick quite firmly to geodesic motion around a black hole. However, there are definitely plunging orbits that will take the orbiting-ant inside the horizon, and there is an innermost stable circular orbit (ISCO) that isn't solid like the yarn: a small perturbation of an orbiting-ant there will knock it into or away from the BH. But an un-knocked ant can circle forever.
The ISCO (3r_{schwarzschild} for a Schwarzschild black hole) is quite a lot of ant-lengths above the horizon of a BH (2r_{schwarzschild}). Spinning black holes have a narrower gap between the ISCO and the point of no return.
The point of no return for a spinning hole is just that: the ant can't backtrack, but will continue moving "forward" from there, and for a massive enough black hole it could do so for an hour or more before it feels the discomfort that precedes spaghettification. The "no drama" conjecture holds that the freely-falling ant won't even notice crossing the point of no return, although astrophysically it is likely to have noticed things falling inwards on different trajectores even above the point of no return (at ISCO around an astrophysical black hole the ant has a good chance of being knocked by something on an intersecting trajectory).
> fabric of spacetime
Misleading terminology. It's not a substance. Spacetime is nothing more than a collection of possible trajectories, and none of them needs to be realized. (Our universe has an enormous number of unrealized trajectories compared to ones on which real bodies move).
> bunched up in a very thin shell
The "thin shell" is just a set of points of no return, and for an astrophysical black hole where exactly each point is can be rather fuzzy since it depends on the outside universe which is filled with moving ants (and galaxies).