Now, when I watched the trailer, about 45 seconds in, the first thing I thought was "Not packing peanuts!" Apparently, in this film, some objects are shipped. This movie would have you believe that museums ship objects in enormous piles of packing peanuts. This is not the case. As in many areas, the way a museum ships objects is much more complex. There are often wooden crates involved, but almost never will there be an object just floating in a sea of peanuts.
Many professional shippers will take an approach called cavity packing (link to .pdf). From the NPS Conserv O Gram:
Cavity packing involves placing an object in successive layers of material (e.g., polyethylene foam) into which an opening is cut. The packing material insulates the object and will absorb vibrations created during shipping. Cavity packing supports the object and cushions it from movement. It also creates a microenvironment for the objects.
So it's a little more complicated. This website has some photos of more typical crating methods. Also, packing peanuts are a pain to clean up and the biodegradable kind make great food for all kinds of creepy crawly pests that you don't want around the collection.
And, for the record, I don't encourage touching ancient sculpture either.