There are a number of ways to categorize safety but I would like to lay out my own personal idea with regard to nuclear power.
First of all, no one is "safe" - there are, at any given time, any number of potential natural disasters that could bring harm to you. For example, being struck by lightning. From this site the odds are around 1 in 200 that your house will be struck by lightning and 1 in 280,000 that you will be. There are similar risks and statistics for flooding, earthquakes, and so.
What's more interesting is that these kinds of odds are odds people will live with and accept every day without much consideration - mostly because the events are relatively unlikely. Though there are many people that live in floodplains, for example, that know they could suffer damage at any point yet they still live there. Similarly with earthquakes, etc.
However, since these types of disasters are natural, i.e., types of disaster than people have little or no control over, as well as unlikely people seem to have little problem with them until the strike.
The next type of "safe" is more interesting.
How "safe" are you in your car?
According to this site if you drive an average amount for 50 years your chances of dying in a care wreck are about 1 in 100. So if I drove 7,500 miles per year x 50 years = 375,000 miles my chances of dying are 1 in 37,500,000. My chances for a dangerous non-fatal accident are much greater.
But the point here is not the numbers - its the idea that when people drive their own cars they feel much more safe - perhaps more safe than they actually are.
During the oil crisis in the 1970's speed limits were reduced from 65 mph to 55 mph. There were also measurably fewer accidents. Yet people wanted the high speed limit because they valued their time more highly than the potential risk the higher speed limit represented.
Next we can look at things like bridges, tunnels and other public works. These all carry a risk of failure and who can say they haven't driven through a tunnel or over a long bridge and felt a twinge of fear? However, these types of structures have been present in human lives for several millennia so people accept the risk with the convenience they bring.
Now let's think about those that create these things: cars, bridges, and so on.
If I make a mistake driving I could kill myself, perhaps a few others.
If I make a mistake designing a car or an airplane the consequences of that mistake can affect far more people - and people, if they don't like the odds my design offers, still have the option to buy a different type of car, travel by train instead of flying, or find another route that does not involve a bridge.
Nuclear power, to me, represents a much different kind of risk - one that people do not normally face.
First of all the consequences of a failure, as we see in Chernobyl, Fukushima, or Three Mile Island have an enormous radius of impact. In the case of Chernobyl a radius of hundreds of miles. Even a bridge failure or airliner crash impacts a very tiny area by comparison. The only thing with an equivalent radius of impact would be a meteor or comet strike.
Second companies and governments are accepting the risk on the behalf of the people. Very few people want the nuclear power plant in their backyard but somehow companies and governments always find a way to get them built. Unlike a highway or bridge no one wants in their backyard the nuclear power plant puts those near it squarely in the radius of impact - whether they like it or not.
Third the impact is long term. If an airplane crashes into the suburbs people die, houses burn. But the next day life goes on - more airplanes don't fall out of the sky each day. In a nuclear accident life cannot go on for years, centuries or even millennia. Its as if airplanes keep crashing day after day for centuries.
Fourth, there is a lot of complex technology involved. Bridges and buildings haven't (and don't) change all that much - Roman's built the Colesseum with concrete - not unlike buildings today. Though humans are new to flight animals have been flying for millions of years - and I can build a hang-glider from simple parts I can buy at hardware stores or lumber yards. Nuclear power, on the other hand, was invented only in the last century - cannot function without computer, electronics, complex and dangerous supply chains for fuel, complex metallurgy, and so on. Each of these involves opportunities for failure and risk on their own as well as contributing the over all chances of failure.
Fifth, there aren't enough nuclear plants in the world to create a proper experience base given the level of objective danger. There are a total of about 500 plants (operating or under construction according to this) in the world. Compare that to 500,000 or so aircraft in the world (a thousand times more) or 50,000,000 cars (a hundred thousand times more). You know a lot more when you do something a a thousand or hundred thousand more times than something else.
So to summarize: Nuclear power is dangerous because
1. The radius of impact is enormous.
2. Risk is not taken on by the actual stakeholders (the population living near by).
3. The effects of disaster are long term.
4. Its overly complex and therefore subject to the significant effects of human error.
5. It hasn't been around long enough nor is there enough of it for people to have a true understanding.
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