Electrical Engineer here with a few comments:
1) A true farraday cage works on both changing currents (AC) and constant currents (DC) effects. The theory is that if you create a hollow space within a perfect conductor, the surface of the conductor will have a uniform electrical potential (Voltage) since any difference in the electrical potential between two points would create an infinite current.
Unfortunately we don't have perfect conductors, we're stuck with metal, so there will always be some potential difference across the conductor, though it will be significantly less than free air. Adding insulation to the inside prevents this potential difference from conducting current through the sensitive electronics inside.
2) For electrical fields, any conductive material will work as a gasketing material, the more conductive the better. Magnetic fields are another story though. While materials like Copper, Tin and Silver are great electrical conductors, their permissivity (ability to conduct magnetic fields) is near that of air. For magnetic fields you want iron, steel, nickle, manganese, etc. Materials with good magnetic properties. If you can find copper clad steel, it's one of the best EM (Electro-Magnetic) gasketing materials availale. If you'd like to maintain close to a water tight seal, pick up some metallic braid (steel if possible) cut the water gasket, slide the braid over the gasket, glue the gasket back together with neoprene cement, then over-lap the braid. The braid may allow some leakage, but it's a lot better than removing the gasket completely.
3) For long-term storage that you don't need to access repeatled, leave the water-proof gasket in place (water is more of an enemy to long-term storage than possiblem EM issues) remove the paint from the cover and base and apply a strip of 1" copper foil tape to the joint. Next, apply a striip of aluminum ducting tape over the copper foil tape (aluminum has slightly better magnetic properties than copper). If you want to be additionally sure, pick up some "Anti-Oxide Grease" this is designed to prevent fire hazzards with alumnum wiring in homes. It is a conductive grease that prevents alumimum wires from oxidizing. Give the water-seal a liberal coat in the grease, then replace the seal. The grease will provide some conduction around the gasket while still providing a water-tight seal.
4) The problem with most cabinats and gun-safes from an EM perspective is they are coated with a protective layer that is not conductive. If you want to turn your safe into a EM shield, pick up some very wide conductive braid (copper braid should be fine, there's a lot of iron in a good safe) then solder or braise the braid to the inside of the full length of the hinge joint. Make sure the braid is wide enough not to interfere with the openning or closing of the safe.
Next, add an EM gasketing material to the entire interface between the body of the safe and the door. One my favorates is a strip of curved "fingers" the mounts to the body and the door compresses the fingers. The best ones are gold plated spring steel, but there are much cheaper ones available (nickle or tin plated copper are the most common)
Finally, mount a very heavy wire to the safe and connect this wire to your water pipes (likely the best ground in your house) to make sure the whole safe is grounded.
Line the safe with neoprene foam rubber padding.
5) The "Skin Effect" was described earlier. AC currents (wall power, lightning strikes, EM pulses) travel on the surface of conductors more easily than on the interior. The "Skin Depth" or the region of a conductor that carries the majority of the current depends on the frequency (higher frequency, lower skin depth) This has lead to copper clad wires (copper layer over aluminum) for lower cost, "lizen" wire - wires made up a dozens to hundreds of very tiny thinly insulated wires woven together and even silver clad typically copper wire with a heavy silver layer on the outside.
While this will help with an ammo-can ferraday cage, the electrical potential fields will be extremely high before you'd even worry about damaging electrical equipment, so the skin-depth effect may be minimal.