The charge of the metal sphere is made up of individual electrons. These electrons are a finite distance apart. If two electrons tried to move to the same location, the would repel each other very strongly and never occupy exactly the same position.

Also, the Pauli exclusion principle forbids two electron in the same state, unless they have opposite spin, but this is a more advance topic from quantum mechanics.

In short, the coulomb law which you correctly cite, applies only to point charges. If you go close to the electrons, this approximation is no longer valid and you have to integrate the charge distribution in space using Gauss' law.

The charge of the metal sphere can be imagined as being made up of individual electrons. These electrons are a finite distance apart and they are not point charges but 'smeared' over a very small but measurable volume. If the surface is defined as going through the center of all the outermost electrons, the electric field will still never be infinite, because the electrons are not point charges but best imagined as charge distributions, little clouds.