If your water is distilled properly and you allow the VOCs to vapor out the top instead of recondesing them then you end up with “pure” water. The problem with pure water is it is the greatest solvent in the world. Pure water wants to dissolve minerals and it wants to pull metal out of containers and it wants to absorb CO2 from the air to form carbonic acid. Pure water wants to get back to a natural equilibrium. What does that mean if you drink pure distilled water? Well it likely wants to pull minerals out of your body and reverse the osmotic potential around the cells it comes in contact with (before it comes to equilibrium).
I think it is probably best to find a mountain spring that is far away from any agricultural activity and that is tested to be clean (meaning no natural uranium deposits etc).
Here is an article that inspired this rant: http://yourorganicgardeningblog.com/who-says-distilled-water-is-safe-to-drink/