It's not possible.
Normal and Electric each only have one weakness. Together, Fighting and Ground cover 8 types, including those two. Fighting and Ground are also the only two types to cover 5 types each, and both of them cover Rock and Steel. This leaves 10 types left, two of which are Fairy and Ghost.
Fairy is only covered by Steel and Poison. Steel covers 3 types, one of which overlapping with Fighting and another overlapping with Fighting and Ground, meaning Steel would only add Fairy coverage. Poison would add Grass and Fairy coverage, bringing the total so far to 10 types.
With that it already seems impossible, but there's more. Ghost is only weak to Ghost and Dark, both of which, only cover Ghost and Psychic. So Fighting, Ground, Poison, and Ghost or Dark would only cover 12 pure types.
Now I did look into Flying Press. Flying Press covers 5 types like other Fighting type moves, but instead of Rock and Steel, it covers Grass and Fighting. So with Flying Press and Earthquake, you would have 10 types. There are multiple types that cover 4 types by themselves, but none of them cover Fairy or Ghost. So even with Flying Press, it's still impossible.
