Thin Cars and Parking Lots

A quick question I came across today:

Suppose you had an infinitely thin car. How big do you have to build a parking lot for the car to be able to turn 180 degrees in the parking lot? 

Think about it for a while, and then highlight the area below to see the answer.

Surprisingly, there is no lower limit on the size of the parking lot. In other words, the parking lot can be as small as you want.                                                                                                

This is better known as the (thinly disguised) Kakeya Needle Problem. Unfortunately, the proof relies on existence theorems which (as far as I know) do not provide an explicit way to show us how we can jiggle the car around until it rotates 180 degrees. So while it might cool to tell your friend you can turn an infinitely thin car on a parking lot with zero area, I know of no way that you can physically show how it can be done.