The Mobius strip is by definition a two-dimensional surface. It's like the surface of the donut in that sense - it has no depth, only width and length. It's not orientable because you can't define an inside or an outside - it only has one surface.
A little more technically, imagine making a little forest of cactus spines or arrows which point up away from the surface on each side of it. If we were talking about the donut surface, there would be no problem - the arrows on the outside point out, and the arrows on the inside point in (into the donut). Now try to do that for a Mobius strip, or this Klein bottle:
http://www.ima.umn.edu/2005-2006/gallery/polthier/kleinBottleNormalShowStill_med.jpg
See the difference?