Answer
Both of them win the bet.
Work Step by Step
Garageman wins the bet because according to the garage frame of reference, the limo is inside the garage with both doors shut for a time of $~~1.36\times 10^{-8}~s$
Carman wins the bet, because according to the limo frame of reference, the limo is never in the garage with both doors shut.
Therefore, both of them win the bet.
