Perhaps to clarify/expand on remarks you made about radiation dose received on a plane, it can be as high as 3-3.5 µSv/hr (30 to 35 background level radiation) and increases with altitude. Still, it's important to notice the 'per hour' bit before jumping to conclusions. Thirty hours spent at 3.0 µSv/hr is a rough equivalent of a single chest x-ray (absorbed dose of 100 µSv or 0.1 mSv) or around ten days of simply living in your normal background radiation.
As far as ground and life being radioactive within the exclusion zone, it's because of bioaccumulation. During the Chernobyl disaster, three main radioisotopes were released. Iodine 131 (halflife of 8 days), Caesium 137 (halflife of 30 years) and Strontium 90 (halflife of 29 years). Iodine decayed so fast that today we might as well assume it's no longer present (there's only about one ten-millionth of the initial amount left after six months). Caesium and Strontium, however, are alkali metals. They eagerly react with water and other common compounds to create products which are both bioavailable and easily soluble in water. The more dangerous of the two is Strontium, which shares many chemical properties with Calcium, meaning it can easily bind to bones and subtly influence metabolic patterns (which I'm unqualified to discuss, simply mention them for someone else to pick up, should they wish to) decaying from within your body. And since nobody can function without alkali metals, they tend to accumulate with time, as you said in your amazing video.