there is more than 3-4 inches under most couches if you start going up into the spaces below the cushions. You need to get creative to get around all the bracing, but it is not too bad. You could probably get 4-5 cubic feet pretty easy, and without raising the couch.
Approach it by building the box to fit the tallest parts under the cushions...but be sure to leave space between the cusion and the top of the box for when you sit. Once the box is built designed, you start cutting out where the braces go and then start adding u-shaped troughs to seal up the areas that you cut out for the braces...
Here look:

Approach it by building the box to fit the tallest parts under the cushions...but be sure to leave space between the cusion and the top of the box for when you sit. Once the box is built designed, you start cutting out where the braces go and then start adding u-shaped troughs to seal up the areas that you cut out for the braces...
Here look: