First of all, length contraction for 0.8c is only 0.6, so not quite the factor two you said. But anyway, let's assume a speed of about 0.87c, so the length contraction is slightly over 2.
As long as the car is moving, the two observers will disagree on the length of the car and the garage, but this will be solved by the fact that they also disagree on simultaneity.
The observer in the garage will see a 10 ft car go into the 10 ft garage, so that its front fender will just touch the rear wall simultaneously with the rear fender passing the garage door.
The observer in the car will see his 20 ft car approach the 5 ft garage, and his front fender will touch the rear wall well before his rear end eventually gets into the garage.
You have to realize there's a bit missing here: what happens after the car hits the wall?
Solution 1: the car, given its speed, will go straight through the wall. In that case, no problem, both observers agree. One will say the front breaks the wall as the rear is entering the garage, the other says the front broke through the wall while the rear was still outside, but that's not really a big deal. Time and simultaneity are very subjective things, and there's no real contradiction
Solution 2: The car comes to a complete stop inside the garage somehow (for example with a series of very strong brakes attached to the garage that grip the sides of the car). Still from the garage point of view, all the brakes act simultaneously and will have to continue to exert a significant amount of force after the car is stopped, since it will want to return to its normal length. The stress may even break the car. Now from the car's point of view, the first break grips the front of the car while the back hasn't entered the garage yet. The rear end keeps moving forward as the next brake grips the car, then the next, etc... until finally the last brake grips the rear end of the car just as it enters the garage. The car was contracted because different parts of it were slowed down at different times. Fortunately the garage expanded during the braking, so that the stress on the car turns out to be the same after all.
Other solutions are possible, but as long as you define the exact transition consistently in one frame of reference, you will find a perfectly consistent view from the other frame of reference. Make sure you take the speed of light into account, though: if, for example, you would say the car stops because it hit the rear wall, you have to take into account the fact that the rear will not immediately know this because information cannot travel faster than light. It will inevitably contract quite a bit more and end up shorter than the garage from any point of view. That's why I used a system of brakes attached to the garage, which can be activated with any timing as required.
