Why can you make two repelling positively charged rods touch? Shouldn't the Coulomb force become infinite?

For a physics lab on the Triboelectric effect, we rubbed two rods with fur which gave both of them a positive charge.

We then brought them close together, and they obviously repelled. We then held one rod down firmly and touched the rods together. We were able to do so.

Why?

Coulomb's Law says that as the distance between two like charges become zero, the force becomes infinite. Using superposition on all the charges in the rod, shouldn't the infinite force have prevented us from touching the rods together?

If the rods were really far apart then the positive charge would be equally distributed throughout each rod. If you push the rods together then the new equilibrium involves fewer charges bunched around the closer points and more of them at the far ends; the energy you exert is the energy it takes to move these charges around. Even if we didn't consider the atomic nature of "contact" and pretended we had a continuous material, you would just find that the point of contact had zero positive charge on it, with the positive charge instead being distributed more toward the far ends.

Presumably if the charges were literally totally immobile and these were literally continuous and smooth materials then you couldn't make them touch for the reasons you stated. But in reality:

1. the charges all have some ability to move (even if it's just on atomic distances)
2. They don't actually come within zero distance from each other
3. The surface is bumpy on a microscopic level so there are probably extremely few actual contact points
4. You're unlikely to actually be touching two charged atoms together given the above and the sheer number of atoms in the material
5. Even if you did manage to precisely line them up to do 4, all you'd have to do is overcome the force it would take for the atomic lattice to rearrange and move the positive charges out of the way.

Also the force on two electrons a nanometer apart is given by $\frac{ke^2}{(10^{-9})^2}=2.309\cdot 10^{-10} N$. So even if you did manage to circumvent all of the above, you'd probably only have a few contact points at most each contributing a fraction of a nanonewton. Due to the incredibly small charge of the electron it would take a lot more than that to generate a significant force.

The reason is that Coulomb's law is only directly valid for point charges, i.e. for charges, sizes of which are much smaller than distance between them. For particular symmetry reasons it appears to also be applicable to spherically symmetric balls of charge - but note that the distance you should take is not between the surfaces - it's between the centers of the balls. Thus, even if you make the balls touch, distance between their centers will still be twice the radius, so the Coulomb force will be finite.

Your objects are not even balls. The actual Coulomb force is a bit trickier to compute, and at small distances it will grow not that fast as for spherical charges as you place them closer. Although this does make it a bit easier for you to make the rods touch, their finite size in all directions is the core reason for the force to remain finite, similarly to the case of balls.