"The sky" is a vague term that originated before anyone knew anything about it. It is just as acceptable to say that birds are in "the sky", as much as the Andromeda Galaxy is in "the sky"...even though birds may only be a meter away, and the Andromeda Galaxy is 2.4*10^22 meters away.
Think of "the sky" as our name for an image comprising everything that is above the ground...everything we see when we look up. It is just an image, it isn't an object.
It is usually more acceptable to say "the atmosphere" when you are actually speaking about the gaseous background material surrounding our planet.
Forgetting about Olbers' paradox for a while (look it up if you are interested), the sky is black at night because black is the DEFAULT image of an abscence of light. Black is the abscence of color.
Stars as a whole appear a heck of a lot dimmer than the sun from our observation point, even though the vast majority of naked eye stars are physically bigger and more luminous. The apparently brightest star Sirius observable in our sky is only appearing about a billionth the brightness of the sun. Assume a more typical star to only appear a tenth as bright as Sirius, and estimate about 5000 naked eye stars...and you have an estimated nighttime brightness on a moonless/planetless sky of about a 50 millionth the brighness of the direct sunlight.
At that intensity, your eyes simply aren't sensitive to colors, and can only pick up intensity levels. You have no option to see any color in the sky, unless you focus carefully when seeing some of the brighter colorful stars. Most colored photos of the night sky are taken with long exposures, so that they can pick up the interesting colors and details.
All and all, there is some Raliegh scattering of the light of the night sky, so if your eyes were sensitive to colors, they could pick up a really dark blue background, instead of the really dark gray background that they normally see.