In a neutral conductor if we assume electrons as point charges, the electric field in the space between them cannot be identically zero. This microscopic field may be very weak. What if we were very close to one of electrons? Shouldn't the electric field diverge?
The field lines of the electrons do diverge, but since we are talking about a neutral particle those same field lines converge on the protons. There will be multiple points in the vaccinate of the atom that positive and negative fields exactly cancel to zero but what we think of as being a neutral particle with zero field doesn't really come into effect until we step back to the macro level where the electrons and protons are treated as a dipole who's field falls of rapidly to approximate zero at any appreciable distance.
How should it be “very weak”? Its field strength is immense, against macroscopic samples. Sure, one shouldn’t suppose a Maxwellian E-M fields inside a matter, especially an electric conductor – this microscopic field is uncertain. One hardly can understand it thinking about it as a vector or tensor field in the spacetime; one needs QFT. There is a related discussion at Ambiguity on the notion of potential in electrical circuits?.