forces - Gravitation law paradox for very close objects?

We all know that gravitation force between two small (not heavenly) bodies is negligible. We give a reason that their mass is VERY small. But according to inverse square law, as $r\to 0$, then $F\to \infty$. But in real life we observe that even if we bring two objects very close, no such force is seen.

Why is this so?

Answer

The inverse-square law holds for spherically symmetric objects, but in that case the main problem is that $r$ is the distance between their centers. So "very close" spheres are still quite a bit apart--$r$ would be at least the sum of their radii.

For two spheres of equal density and size just touching each other, the magnitude of the gravitational force between them is

$$F = G\frac{M^2}{(2r)^2} = \frac{4}{9}G\pi^2\rho^2r^4\text{,}$$ which definitely does not go to infinity as $r\to 0$ unless the density $\rho$ is increased, but ordinary matter has densities of only up to $\rho \sim 20\,\mathrm{g/cm^3}$ or so.

Tests of Newton's law for small spheres began with the Cavendish experiment, and this paper has a collection of references to more modern $1/r^2$ tests.