Why doesn't a C4H8 alkane exist? This structure:

     H   H
     |   |
 H - C - C - H
     |   |
 H - C - C - H
     |   |
     H   H


Which would be $\ce{C4H8}$, like butene, but an alkane, because it's functional group is C - C.
It does exist. The formula $\ce{C_{n}H_{2n+2}}$ is only applicable to straight chain alkanes. They can't have any degrees of unsaturation, which is defined as a multiple bond or a ring.