I'm trying to install a ceiling fan. Including the switch wire, all the wires in the box are hot (even the white wires). I'm not sure how this box was wired before but there doesn't seem to be any neutral wires. This doesn't make sense to me. Can a light or ceiling fan be hooked without a neutral?

Why do you think they are ALL hot?

I bet the circuit is still on and you are reading a backfeed.
I'll also ask the same thing, how are you testing this?