What space looks like in space depends on what is around you that might scatter light. The human eye is very adaptable to varying light conditions, in low light conditions the iris dilates to let in as much light as possible, in high light conditions it contracts to restrict light to give the retina a consistent amount of light and to protect it from damage from too much light.
On the Earth you cannot see the stars in the day because the atmosphere scatters light from the sun. The stars are still there of course but they are drowned out by the other light. Also, your pupil is too small to give the sensitivity to low light emitters. On the earth at night how many stars you see depends on the condition of the atmosphere and the amount of scattered light there is, which is why modern observatories are built in high altitudes far from any light pollution.
In space what you see is determined by what is around you and what is scattering light in your direction. The surface of the moon is very bright in sunlight, so if you are standing on the surface looking at the moon your pupil will contract and you will see fewer stars.
If you are floating in space no-where near any light scattering source, with the sun out of your field of view then your eyes would dilate out and be at their maximum sensitivity (after 20 minutes - it takes time for your night vision to kick in fully) what you would see is the deepest, deepest black punctuated by the unwavering pinpricks of stars (no twinkling as there's no atmosphere), and a bright glow from the milky way. It wouldn't be that different from looking up at the sky from the palomar observatory (or any other high point) on a clear and moonless night - at palomar you wouldn't be looking through a layer of glass or plastic designed to keep you from asphyxiating, but the view in space would be even more spectacular.
The moon has no atmosphere to speak of to scatter light, so if you lie on the surface of the moon facing up, and the sun and earth are not in your field of view, it would be the same thing as if you were in open space. As soon as the earth, moon or sun came into your view the light will cause your eye to contract and you'd see less stars.