You need dense armor to protect against ionizing radiation. There is no way around this: the only way to stop gamma rays and xrays is mass. Mass means weight. Massive substances like metal take less volume than less massive substances like bone or chitin. I suspect the radiation protection / unit weight will be comparable regardless of the substance but less massive substances will be very bulky.
Calcium is about the densest stuff one usually encounters as structures in biology. But my search turned up that bees deposit granules of magnetite in their bodies. Maybe for magnetoreception? http://jeb.biologists.org/content/jexbio/180/1/1.full.pdf
Iron is good and dense and biological systems are good at handling iron. Your creature could accumulate iron in its shell which would add mass with less volume. Iodine is twice as dense than iron and biological systems (including ours!) are good at accumulating that too. An iodine fortified shell would be doable.
If the radiation is from one direction only that is easier, because you do not need a circumferential shield. For example if radiation is only from above a creature could have a tortoise-like shield of armor and stay completely below it.
The needed mass to protect will of course increase with the size of the animal you wish to shield. It will be easier to shield a small animal than a large animal. Small to medium tortoises, for example, carry relatively heavy shells compared to giant tortoises, which have porous, lightweight shells. An snail or abalone can easily tote around a shell several times heavier than the entire live creature.
If you want a big creature with a very massive shell / armor a way to still have it be able to move is to have it swim underwater, and have the shell be buoyant.
A nautilus is an example of this.
The shell / armor would be just as massive but by enclosing large enough air chambers it becomes weightless. The only remaining problem is inertia, and so you cannot be in a hurry. I could imagine a nautilus shell type arrangement configured so the animal stays completely inside, propelling itself via the siphon.