At night, I can look up and see the Milky Way across the sky. My question is, supposing our Solar System was, instead of way out on an 'arm' of the galaxy, near the galactic center, would the night sky be much brighter? If so, how close would we have to be to never have darkness of night?
The Wikipedia article on "Stellar density" says the stellar density near the Sun is only 0.14 stars per cubic parsec. It suggests that the density in the central core and in globular clusters is about 500 times as great.
According to the List of nearest stars and brown dwarfs in Wikipedia , there are 61 stars within 15 light years of the Earth. Dividing 61 by the volume of sphere of this radius, we obtain 4.3e-3 stars per cubic light-year, or 0.15 stars per cubic parsec. Thanks to user31264 for providing this information, which is consistent with the information from the previous link.
According to the Wikipedia article on "apparent magnitude", the total integrated magnitude of the night sky as seen from Earth is -6.5. Making that 500 times as bright produces a total magnitude of about -13.2 (5 magnitudes is a factor of 100 in brightness). The maximum brightness of the full Moon is -12.92.
So even with 500 times as many stars in the sky, the total brightness would be only slightly greater than that of a full moon.
(This assumes that the average brightness of the core stars is similar to the average brightness out here in the Galactic suburbs.)
Parts of the core might be even denser than that.
(I've updated this with new information and deleted and old link whose numbers appear to have been incorrect.)
Okay the notion that the sky would be 100 times brighter because there are 100 more stars within a cubic parsec is ludicrous. First it would discount the 100x 100 more stars that would lie within the next parsecs distance from us. More importantly it discounts the far closer proximity of the stars within our Parsec. With 100 stars within our particular cubic parsec odds are one would be just outside of our solar envelope... heck one very possibly would be closer than Planet 9!
Could you imagine if Betelgeuse was a solar month away! The answer is in the galactic bulge you would never need a flashlight but bring SPF1,000,000 !