Why does electrostatic field become zero inside of a conductor?

Yes, an insulator can begin to conduct electricity under highly extreme conditions. Too strong of an external electric field can be applied across an insulator, which can let it pull electrons away from their atoms. That frees them to conduct electricity. This phenomenon is known as dielectric breakdown.

Distilled water is one of the best examples of an electrical insulator. Very few ions are free and they are not enough to carry charge. But, if you look at tapwater or saltwater, it can be an effective conductor, as these liquids have dissolved salts and minerals acting as mobile charge carriers. They allow electricity to flow. 

The best conductor of electricity is Silver (Ag) at room temperature. It is not used in normal wiring even though being the best conductor because of its high cost.

• Electrical Conductivity ($\sigma$): $6.3 \times 10^ {7} \ \text {S/m}$

• Electrical Resistivity ($\rho$): $1.59 \times 10^ {-8} \ \Omega \cdot \text {m}$ (the lowest among all metals)