During the aftermath of Hurricane Katrina the water was extremely dangerous. It was so dangerous in fact, that you weren't even supposed to touch it. Serious water contamination is one of the main problems during a disaster, so how do you treat the water? Here's a link to the American Red Cross document titled "Food and Water in an Emergency" (click here). This document highlights why you can rely on distilled water in these situations and you cannot rely on filters or reverse osmosis systems.
In short, the first step in distillation is to boil the water, which kills biological contaminants. But remember, as you boil water the pure water is leaving as steam so the water that is left behind actually gets more concentrated. Distillation not only boils the water but we capture the pure steam, separate it and cool it back down into liquid water. Thus what you are getting is not just biologically safe, but it has removed virtually all of the other chemicals.
Distillation is by far the best method for purifying water in any situation.