How do you avoid getting radiation in starbound?

You need a Radiation EPP Upgrade:

Its an EPP aka Environmental Protection Pack, you can wear these in your backpack slot to defend you from a variety of hostile environments like radiation, extreme cold, or that fact that there is no breathable air on around you.

To craft this EPP you gonna first need a Breathing EPP which can be crafted at the Anvil using:

• 10 Glass
• 6 Tungsten Bar
• 1 Living Root

After you created your Breathing Upgrade Pack, you can upgrade it at a Forge to a Radiation EPP using:

• Your Breathing EPP
• 4 Copper Bar
• 6 Titanium Bar
• 1 Venom Sample

(This answer applies to the question when it was asked, and was only applicable to the beta version of the game. Paul's answer is correct for the newest version of starbound.)

As you complete missions in the main quest line you gain access to new nanoskin suits. The radiation skin is given when you complete a quest to gather 300 kelp for a fish person on the outpost. Unfortunately this quest does not appear until after you defeat dreadwing. I was able to defeat dreadwing the first time by using some bone swords that I found in a bone biome in combination with the boneboo fruit, which boosts your melee damage.

THIS IS OUTDATED You need the "Radiation Protector" You obtain it after completing the quest "The Grass is Greener"