I wouldn't worry too much about the storage, the odd time I've put the sides in there, they rub together and get marked.
They rattle too!
An inexpensive solution (for those with the storage slot) is to repurpose a couple of the wheel-bags supplied with decent bicycle racing wheels. These have a zip, are slightly padded and are the right size. If you are not a bicyclist, ask a friend who is they are likely to have loads of the things left over in their constant quest for faster wheels.
