That sounds like a good idea April and Andrea I don't think above ground water tanks are too expensive to buy. Maybe a few hundred for a small one?
We have tank water at our house Only because we do not get access to town water where we live! The water is always perfect for the fishies and it tastes better too lol.
Our water does go through a filter and stuff before it comes into the house but I think this is mainly to remove any bugs/dirt that has got into the water.

The only problem might be that the water in the tank could freeze in winter...?

Not if you get a rain water tank Nossie, it would be separate to your old drains system so I don't see where the chemicals could come from. I was suggesting a prober rain water tank, not what April said she may do as that may be a bit unsafe  The air well I guess that is true but wouldn't the air get in tap water too?? I've done alot of research on rain water due to a plant I have which can only have rain water and if humans can drink rain water with no treatment then I'm sure it is pretty safe.

I don't actually know with fish though and I would also put alot of research into it but in April's case I would definitely consider it.

My rain water runs straight from the gutters on our roof into the rain water tanks which are underground. Once the water is in there nothing can get into the tank. Before the water comes out of the taps into our house it runs through a filtration system.
The only way I can see rain water being a problem is if you lived in an area with high pollution or acid rain or something...other then that if you set the tank up properly then I think it is a great idea

I have had fish swimming in it for 5 years and as far as I am aware they have had no problems due to the quality of tap water. The parameters are perfect and do not fluctuate