Does perfect insulation exist?
Insulation \$\rho\$, breakdown voltage \$U_{br}\$ and dielectric permittivity \$\varepsilon\$
Insulation is the ability of a material to block the flow of current through it, so that means that this material has a high resistivity. It is measured in \$\Omega m\$(Ohm-meter).
Breakdown on the other hand is a process in insulators, it can be quantified in terms of a breakdown voltage (for specific frequency, temperature etc.). That means that if you pass a specific voltage the material loses its insulating ability and fails catastrophically(melting, arcing, etc.). You can also measure it as \$E_{br}\$, or breakdown field it is measured in \$\frac{kV}{mm}\$(kilovolt per millimeter), air has \$3\frac{kV}{mm}\$ for example.
The dielectric permittivity is yet another property of insulating materials. The relative permittivity \$\varepsilon_r\$ tells you how many times the electric field will be lower in the material in comparison with vacuum. Say if you had a capacitor with vacuum as dielectric, and another one with some material with a specific \$\varepsilon_r\$, the electric field in the (with everything else the same) in the second capacitor will be \$\varepsilon_r\$ times smaller than in the first one.
Superinsulators
Similar to superconductors, there are superinsulators. Superconductors will be superconducting only for some specific circumstances. That means specific temperature, magnetic field and current density. You can't put infinite current through it, as at some point it will lose its superconducting ability.
For superinsulators you have a similar story, they won't let any current pass until you keep the parameters (voltage, temperature) within specific boundaries. They will therefore lose their superinsulating property at one point.
Vacuum
Because of the discussion on Majenkos answer I'll have to talk about this, too.
How can vacuum not be a perfect insulator? It has no current carrying ability as it has no particles that could conduct, so there can be no current flow. Therefore the conductivity should be zero, isn't it? Well there are always very small amounts of particles that will create some conductivity, and there can be breakdown processes in the vacuum.
Look at high voltage switchgear. There are vacuum switches which go (as far as I know) up to 160kV. That means that they can handle the separation of two contacts that have this kind of voltage between them. During the switching operation an electric arc is created. This arc is mainly fed by the let's say evaporated contact materials. So the vacuum will conduct only because of foreign particles.
There always have to be two electrodes at a voltage difference for conduction, if vacuum is in between them we just need a big enough voltage difference so that particles start to move from one electrode to the other creating a conducting path. So eventually everything will conduct because of the electrode material.

Image of a high voltage switch, source for image: Mitsubishi Electric