Blood is already just about as good a radiation shield as you're going to get. That's because it's mostly water, and water is a great shield.
Just like a bulletproof vest, the two main properties that you're going to want in a radiation shield are,
- Being there.
- Safely absorbing the energy of a collision.
It turns out that there are plenty of materials that are relatively safe when irradiated. (Choose the wrong one and your shielding is a breeder reactor.) Lead and water are two great examples. So, in stark contrast with the challenges involved with making bulletproof vests, the biggest problem you're going to have when designing a radiation shield is arranging things so that the sheild is there when a high-energy particle passes by.
I'll go straight to talking about neutron shielding because that's the toughest case.
As you may have heard, atoms are mostly empty space. Neutrons are neutral (meaning that overall they don't interact with electric fields), and they will only ever involve themselves with other matter through the weak and strong nuclear forces. Those forces are very, very short-ranged, and for all practical purposes only act over length scales around the size of a nucleus.
As a result, your neutron shield's biggest concern will be putting enough nuclei in the path of the radiation that the chance of it hitting the shield becomes larger than its chance of hitting the animal. You'll want to do this with relatively high-density materials. Once again, water and lead.
If you're willing to invent new materials, then your animal could have a shell of pure nuclear matter (like a neutron star.) It's strongly advised to put the material in a shell or on the animal's skin, because if its in the blood it will waste a lot of time being pumped around through internal organs that aren't supposed to have any radiation nearby in the first place.
Charged particles (like electrons, protons and helium nuclei (alpha particles)) can be deflected by magnetic fields. If you're willing to take the hits from neutrons and photons, your creature can protect itself from the charged particles whizzing through space by filling itself with very powerful magnets. I could imagine some tiny blood-suspended superconducting crystals accomplishing this, although it might be hard to stop them from sticking together (clotting, if they're in blood) like magnets tend to do.