DNA repair mechanisms are well known, and taught in any introductory-level genetics course. The BRCA system is very well studied due to its clinical significance. The lowest yearly dose of radiation that has been shown to have an effect is 100 mSv, which is four orders of magnitude higher than we're talking about.
Radiation safety authorities have a long reputation of overreacting to low level radiation sources. For example, items that went into a radioactive area, and have measurable lingering radiation, were once (and perhaps still are) classified as radioactive waste. Even if such items were just as or even less radioactive than they were before being exposed. I can't entirely blame them, the radiation poisoning after WWII in Japan was an unexpected tragedy that we hope to prevent in the future, but we know a lot more about radiation now.
That said, linear no-threshold models would predict that everyone should show symptoms of radiation poisoning around 80 years of age, and people in high altitudes long before that. It's not a defensible hypothesis, having no supporting experimental evidence or physiologic rationale. Safety estimates are very conservative, so that's why they assume zero DNA repair, but that's not being realistic. The linear models are a simplification that breakdown at the extremes.
Of course, setting the exact lower threshold is quite difficult, since the population varies tremendously, plus it's somewhat trivial to differentiate between zero and essentially zero. The only time it becomes relevant is when you multiply very small, incorrect numbers (i.e. probability of causing cancer at trivial radiation doses) by very big numbers (i.e. millions of people) and get a number greater than zero.
You can visualize it like blood loss. If a person loses 2 liters of blood, they'll show certain symptoms. If they lose 3, the symptoms are a lot worse, and they'll probably die if they lose 5 or 6. You could likely construct a no threshold linear relationship out of this. But, obviously it breaks down when you say that you'd expect one person to die from blood loss for every 5,000 paper cuts.