I have a large trash barrel placed underneath one of my gutters, so that they water streams down right into the barrel in the summer. It's placed close to my garden, so I don't have to lug the water very far.
It is always full. I water my plants every day in the summer, and I've never had a day without free water.

No, the water doesn't get icky because there's so much water going in, that it displaces the old water by force.

I love this system and highly recommend it!