I've looked through pretty much every combination possible, and I'm pretty sure that there is no "clean" sets of five. That is: there is no set where every type does normal damage against all the types in the set other than the ones it's weak or resistant to. There's always one type that can hurt two types in the set, or one that's immune to one further down the set etc.
Pokemon is an awesomely complex game. It's no surprise that there's no sets of five. I found a few sets of four though!