I've had great success (with Ikea furniture) replacing drawer bottoms and cabinet backs with thin ply. From memory Homebase had some which was just a tad thicker than the standard issue hardboard, which was great, because although it required a lot of force to get it into the slots in the drawer sides etc, once in it was very rigid. And of course all the sheds will cut it to size for you - just measure the original bits of hardboard and Robert is your father's brother.
The work doesn't stop there, of course, because you have to sand and varnish or paint the plywood, so it adds quite a few days to the elapsed time for assembly, but it's amazing how much rigidity it adds to the standard product. Gluing all the joints helps too.