Well these long term health effects are quite tricky to quantify.
Even a perfectly healthy person that lived in some sort of shield to protect them from all natural radiation could still get cancer. Things like UV light or carcinogenic chemicals can also damage DNA. Even if you are not exposed to those, still the processes inside living cells are not 100% reliable. There is always a tiny chance that something might damage it(like oxidants), or that the DNA does not copy perfectly when a cell divides. Because of this cells have repair mechanisms that constantly fix damage to it, but these are also not 100% reliable, especially if there is too much damage. But even then when "bits get flipped" in DNA that will in a lot of cases do nothing, or cause a cell to do something too radically different and die, its only when the DNA changes in exactly the right way to cause the cell to successfully uncontrollably replicate is when you get cancer.
Ionizing radiation is just one method of many that can "flip a bit" in DNA. So technically yes no amount of ionizing radiation is safe, but below certain levels its contribution to DNA damage is going to be so small that the other DNA damaging methods overshadow it.
And since the effect of damaged DNA is the cell doing the wrong thing or dying all together (As well as radiation disturbing other cell processes) is why radiation sickness happens after short exposure to intense radiation. The cells have to work hard to repair the damage and a lot of dying cells around puts a heavy strain on the body. There is a chance a cell turned to cancer and survived, but if not while the person survives the massive surge in cell death, then they can recover perfectly fine. Just that the more radiation they got the more the odds are stacked against them.