Actually, take a look at a car. You might find it is standing on four round rubber things called 'tyres'. As most things made of rubber, they're not very good to use if you want to ground something as they're non-conductive. IOW, the car isn't all that different from the aircraft...
The sheet metal in the skin forms a Faraday cage though, so the interior will be rather well protected from the strike just like FBU described.
However, a bad strike might just end up damaging the skin in the entry or exit points. Having a hole blown in the radome isn't all that uncommon. If you're unlucky, a strike can take out avionics on a large scale. They're well shielded and tested but there is a lot of hard-to-control energy in a thunderbolt. :/
I thought I was doing good trying to avoid those airport hotels... and look at me now.