Radiation levels at Chernobyl are at fairly "safe" levels. I'm not saying build a vacation home there, but short-term exposure is fine. The area has been slightly more "opened up" in recent years with more and more researchers spending time there studying the effects of urban decay, the impact of radiation on the environment and there's even occasional tour groups who go through there. Those who "work" there long term usually do so in two-weeks on, two-weeks off shifts or something like that. And I'm sure are taking iodine pills or something. The more dangerous areas are closer to the reactor that melted down where very, very limited exposure needs to happen. But even then, there's still people who work there, granted in protective suits.
Besides regardless of the "reality" in the movie we're told the radiation levels were safe. Hell even if the pool they fell in was a cooling pool for used reactants they would have been safe unless they actually swam down and got close to the reactor casks.