My carriages(I have 4(edited to say-miscounted, it's 5; one is 'big horse' sized) two wheelers, 3 four wheelers)are all stored in my insulated barn, under cover(I have swallows in the summer, and dust all year round!)! The one Jerald I still have is in its Jerald-made cover, two of the four wheelers have a 'drop on' cover, the others are covered with old sheets and, on top of that, plastic sheets or those blue poly tarps. A well-ventilated area is importanct, and, it is a good idea not to cover one directly with plastic or poly; if there is any humidity, especially if in a small, close area, the cover may 'sweat' underneath, and can damage the finish of either a wooden OR a metal cart. Some two wheelers can be stored with the shafts in the air; it kind of depends on what it 'rests on' when placed in that position. I always make sure that I place a short piece of 2x4, or an old saddle pad, or some such, under where the cart will 'rest' when placed vertically.
Margo