Singularity in Rutherford cross section
The problem with Bob's answer is that you still have a singularity in the scattering with a bare nucleon. This singularity is real.
What you have to remember is that the Rutherford crosssection is a theoretical calculation. Like all calculations, how well it matches the real world depends on how well the assumptions match the real world.
The assumption that matters is that the interaction between the scattered particle and the scattering particle is instantaneous and depends pretty much only on the closest point of approach of the particles. This is a good approximation for wide angle scatters: a free particle comes in close, has a sudden interaction, and then a free particles moves out.
For interactions where the two particles never get close, this is a poor approximation  and that's exactly where the Rutherford crosssection becomes singular. So the lesson you should take from the singularity is not that you can ignore it because it's difficult to prepare a system that's not under the influence of nearby electrons: it's that you should recognize that this is telling you that (at least) one of the assumptions in the derivation is no longer valid.
