There has been lots of talk about doing experiments to see whether gravity follows an inverse square square law at very small distances. But what about the electrostatic force?
Is there any direct experimental evidence that the electrostatic force follows an inverse square law at distances of a few femtometers?
When you're probing distances that small things get messy. Electron anti-electron pairs, and many other matter anti-matter pairs, pop in and out of existence. This leads to a process known as vacuum polarization that causes the 'bare' charge to be screened, appearing smaller when you're farther away until the effect drops off (I'd guess at around a distance of $ h/(m_e c) \approx 2 \times 10^{-12} \operatorname{m}$) and to alter the observed force between particles.
In terms of the "bare particle" calculations that go into computing the final force, though, the behavior is consistent with photons being massless and space-time being continuous. Those two facts, combined with a two derivative limit on theories and $3$ dimensions of space, imply a $1/r^2$ force for the bare particles. All of these statements have been confirmed to the energy levels and length scales reachable by particle accelerators. For the LHC that is at an energy with order of magnitude $1 \operatorname{TeV}$ which is equivalent to a length of $10^{-21}\mathrm{m}$.
I think this question involves so called extra-large dimensions. The idea here is that compactified dimensions of $10$ and $11$ dimension supergravity and string theory might appear on some scale comparable to accelerator physics scales. A field that appears as $E~=~q/r^2$ occurs because the source of the field can be enclosed with a two dimensional Gaussian surface. The field vectors pierce the surface with area $4\pi r^2$ for a sphere. This leads to the celebrated Gauss law $$ \int_{\sigma=\partial V}\vec E\cdot d\vec a~=\int_V\nabla\cdot\vec E dV~=~4\pi q, $$ where the Gaussian surface around a charge gives a measure or accounting of that charge, or a total number of charges in the region enclosed.
If there are these extra large dimensions, say $N$ of them, that "unwrap" at $TeV$ energy scales or something similar then the appropriate Gaussian surface to isolate the charge is $N-1$ dimensional. For most compactification schemes $N~=~6$. The measurement of fields on very small regions would lead to disagreements with a $2$ dimensional Gaussian surface, which would in principle be detectable. It would be as if field lines that would ordinarily define a conserved charge are being lost in these extra dimensions.
There was a lot of interest in this early last decade. I have not heard that much about the status of this research. If a measurement of this had been performed I would have expect to hear the trumpets sounding, and so far nothing.
