The anechoic chamber in the Orfield labs has been recognized as the quietest ever recorded. However, if you are talking of naturally occurring quiet then probably some of the deep cave systems with no water content are as close as you will find.If you insist on being above ground it is harder to pick. Since any birds, animals, water or wind will rule the site out I would suggest some deep, steep-sides valley in an arid part of the world might come close. If you can get rid of tourists then Petra might be a good start

Some desert situations at various times of suitable days can be delightfully quiet. A few years ago I was privileged to visit Vaalputs, a nuclear waste repository in South Africa. One of my companions said in effect: "The silence hits one like a blast."

With no wind and nothing audible for miles, it was lovely. I have never been in a place too quiet for me.

Technically Mike is correct about anechoic rooms etc, but there is a sort of emotional stillness about the open air with not even a grasshopper stridulating...