It's true that radiation is more damaging to cells when they can't engage in repairs. But damage is nothing to worry about in this case. When e.g. a gamma ray photon breaks a protein molecule, that molecule is rendered nonfunctional; enough such events will kill a cell. But in the context of cryonics, a broken molecule is as good as an intact one provided it's still recognizable. Rendering it impossible to tell what the original molecule was, would take far more thorough destruction.

From Wikipedia, "The worldwide average background dose for a human being is about 2.4 millisievert (mSv) per year." Even a lethal prompt dose is a couple of thousand times this quantity. And you can take maybe 10 times the lethal dose and still be conscious for a little while. So that's 20,000 years of background radiation verified to not even significantly damage, let alone erase, the information in the brain. I'd be surprised if the timescale to information theoretic death by that mechanism was very much less than a billion years.