Its a popular discussion point
I think it largely depends on how long you are going to be storing them for. If its a few weeks then you dont need to be worried about being too careful as its not long enough for most problems to develop. However if its longer then you may want to take time and some extra precautions.
Make sure not to store dark/bright coloured ponies pressing against white or light coloured ponies - it is possible for colour to stain. Personally I prefer to wrap ponies in individual white acid free white tissue paper or open [or air punched] zip lock bags. Wrapping them prevents any stains from other ponies or paint from sticking or them from sticking to the plastic in the box.
I try not to store more than 2 or 3 layers of ponies on top of one another - legs can get pushed out of position rather easily and simply through the weight of ponies on top of one another. I prefer stacking ponies in standing position as it normally prevents any warping issues developing.
I am not fussed on plastic or cardboard boxes [unless there is even the remotest possibility of a box getting damp and then its plastic]. I do however like there to be air vents in the box. Plastic sweats and lots of plastic in one place can cause a problem - so best to allow it to breathe.
Good luck and have fun restyling your pony space