there are two "types" of radioactivity, really. there is the energy part, and that energy can be very high and cause burns or death (if enough exposure). There is the particle part, which can cause collision with molecules and destroy molecules, so there is still destruction and possible death of cells, but also non-fatal events that have possible genetic changes (the building code gets broken and won't work good).
Either way, the main issue is that you cannot tell that you are being exposed while it is happening, at least not from your own senses, so you can be severely or fatally harmed before you ever know it.
As with pretty well any hazard, there are different degrees of severity that are possible. Minor sunburn is not really a major issue even if it is solar "radiation" that causes it. But it can also involve genetic changes that you cannot know about until you get a cancer on your skin, maybe years later.
The "danger" is that energy and particles are out there, and can be very extremely intense, and you cannot know it until it is too late, unless you are aware that there is a radiation source and take steps to minimize its effects. Kind of like carbon monoxide in a way: you can't tell it is there so you can die before you know what is happening. Pretty scary thought.