It was a 3/4 size model, wasn’t it? Surely they could tow it on a trailer?

Edit: from wikipedia

Art director Matt Jefferies originally envisioned a sleek, streamlined shuttle based upon his background as a pilot. The curved shape proved too expensive to build for the first episodes.[3] AMT offered to build a full-sized shuttlecraft at no cost in exchange for rights to market a model kit. The final design of the mockup, by Gene Winfield,[4] is 24 feet (7.2 m) long and weighs one ton, has a plywood hull, and was built in two months by a team of 12 people. A separate set was used for interior scenes as the mockup was too small for filming.[5]