In almost every problem in electromagnetics, we will replace a good conductor, like copper or alumin(i)um with a perfect electrical conductor (PEC). A PEC has infinite conductivity and infinite number of valence charges. So for an electrostatic problem, it means we assume that no amount of electric field will ever exhaust the supply of electrons in the metal. For electromagnetic problems, we assume that the excited currents on the metal are confined to the surface and perfectly cancel out the tangential electric fields at the surface (and the normal magnetic field too).
For electrostatics, it's an approximation that we can make without much error. For electromagnetics, it depends on the frequency of the problem but generally anything on the order of megahertz and up will be fine.
The charges in a metal, say copper, arise from the fact that the valence electrons will move into the conduction bands just through thermal excitation (each Cu atom has one valence electron). So whenever you apply an electric field in a metal, the valence charges will move in response due to the Lorentz force. The Lorentz force will move the charges throughout the metal, as the charges move away, they leave behind positive ions of the stripped copper atoms. So you get a charge separation, immobile positive ions on one side and the negative electrons on the other side carried away by the applied electric field. The electrons will keep moving until they reach a the boundary of the metal. Once at the surface, they collect, but as they collect they will create a secondary electric field inside the metal between themselves and the stripped ions on the other side. This electric field opposes the applied field and will continue to build up as more electrons get stripped and move until the two fields cancel eachother out. At this point, with no net field, and barring thermal movement, the electrons will remain stationary since there will be no force acting on them.
In electrostatics, we estimate this behavior by having the appropriate density of negative and positive charges move to the conductor's surface. So the question becomes, at what point do we deviate from this model? Well, if we are to ensure that the ions will always move to the surface, then we can only strip the valence electrons of the atoms on the boundary of the conductor. This is because the positive ions are immobile, they are bound by the bulk lattice and so if we start to strip electrons from the inner ions, then they will lie at some point in the interior of the bulk which violates our PEC estimated behavior. So a rough calculation shows that for a 1 m thick slab of copper, the maximum electric field that we can use that will still follow the PEC approximation is on the order of 10^11 V/m. That's friggin huge.
Now, if we relax our conditions and just want to find what the maximum electric field we can apply that will strip all of the valence electrons, we can get a vague idea by allowing the stripped ions in the copper lattice to move instead of being immobile. We could solve the problem with them being held in place, ignoring the changes to the lattice constants, but that is a harder problem to solve. In this case, we will move all the valence electrons to one side of the conductor and the stripped ions to the other side and find that the electric field is on the order of one zettavolt per meter (SI units are fun).
Your question about would the electrons be left behind, no. There is nothing holding the valence electrons to the atom. In the crystal, the bond is so weak that the normal room temperature thermal energy is enough to excite the electron from the valence to conduction bands (remember the "sea of electrons" model for metals?). At equilibrium however, the electrons do not move around in a way that creates a net electric field or net local charge density because the coulombic forces of the system will always oppose such build ups. Only when we introduce an external field will we achieve a net movement of charges. But the movement is not restricted, as long as we have a net electric field, the charges will move in response to it. So if we have a large enough electric field that will over power the largest opposing field that the bulk's electrons can set up, then we can no longer use the PEC electrostatic model because there will be a nonzero electric field inside the conductor.
EDIT: Paperdoc's point is that due to the astronomical electric field that you need to exhaust the charges of a good conductor, you will reach the dielectric breakdown voltage of your dielectric before that can occur. At that point, the dielectic will allow a conducting path between the two points of opposite charge, allowing the charges to flow and redistribute themselves. The end result is that there will no longer be a net charge or a greatly reduced net charge.
Basically, the main point here is that the electric fields/charges needed to achieve this are not something that can be physically obtained or maintained statically in a system.
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