Why are people so afraid of nuclear energy?

Everyone have seen footage of nuclear bomb explosions and that instill fear in all people. And for those who think deeper about nuclear energy, there are two issues.

First, there is fear of radioactivity in case of meltdown. It happened in Chernobyl. But, nobody builds any more such plants without protective dome. However, in case of power plant with protective dome (as they are all new ones), risk of large escape of radioactivity is negligent even in case of meltdown. It is not well known to public, but properly built automation makes such event nearly impossible in any case. Of course, it is never possible to achieve statistical zero, like in many other cases.

Second, there is issue of nuclear waste. It stays radioactive and potentially lethal not for millions, but billions of years. And some elements like plutonium and polonium, which are result of nuclear reactions, are lot more poisonous then uranium. Problem with waste can be solved by building safe storage in sparsely inhabited, seismically quiet areas. But, here kicks in NIMBY (Not In My Back Yard). Carbon dioxide from burning coal and oil is at least spread evenly over whole planet...