Image Credit: STSci/NASA
If we were floating around in intergalactic space there would be virtual no illumination and the sky would appear completely dark. With the naked eye, you may be able to distinguish 2 or 3 faint patches of light interrupting the blandness of the sky (these would be nearby large galaxies). Keep in mind from that from earth's vantage point in the "Local Group" of galaxies only 3 galaxies external to our own milky way can be seen with the naked eye. It would be quite an unnerving experience IMO to be floating around in intergalactic space!
Interstellar space is a bit more interesting. As a fun exercise I integrated the brightness of all the stars in the Tycho-2 star catalog (2.5 million stars) and derived a brightness that was equivalent to a thin crescent moon. This is still very little ambient light, and I doubt you could see anything more than silhouette's of objects against the sky background.
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