Yakima has a new "FatCat" ski clamp that is very low profile. However, you might need to lift them at an angle to provide clearance (between roof and binding) for skis with lifter bindings, and this may cancel out some of the height benefit. But that's one of the lowest profile ski racks I know of.
You would get the lowest possible overall height by getting mighty mount clips made specifically to attach the FatCats to your factory crossbars (assuming you have them). Any other crossbar arrangement will be taller I am sure.
14" actually sounds like a lot to me, so other rack choices might be OK. Just make sure the overall height with skis fits under the door!! Plenty of us have cars and racks that fit in the garage OK, it's the cargo/loads that are the problem!