I don't think they make a 4 drive bay for 2.5" drives, especially external ones but you can make something from other parts. Keep in mind that it may prevent damage to the case, but a fall can still damage the platters and internals, especially if they are moving. Keep in mind there can be a heat issue is you fully stick the drives in foam or something.
Best thing to do, get a NAS box, hook it up to your network and access the files over the network where the box sits on the floor or desk or somewhere without being moved.