abucted humans/time scale
from trivia
The movie appears to take place in the late 1950's/early 1960's. Cars and clothing appear almost exclusively from that period. Also, in the flashback of Dr. Schreber being forced to erase his own memory he's seen wearing an old-style medical smock favored by doctors of the period. This reinforces the idea that he was a kidnapped psychiatrist being used by the aliens to manipulate human memory.
judging by the city being based on the inhabitants memories its evident they were from 1940s-1950s America
as to when the actual film is set (in terms of the time era on earth during the film not taking into account potential light years etc) it cant really be more than the 1960s as Kiefer hasn't aged that much from when we see him severely beaten erasing his memories in the flashback, humans that die in Dark City are then reanimated by the aliens as vessels and there appears to be hundreds of them - so maybe if the aliens abducted a bunch of people throughout the 40s/50s then it could be about 10 years on (which would account for so many dead?)
as to where they are in relation to earth, it looks set in deep space maybe light years from earth in which case time has no meaning (a very eerie notion of this city experiment/vessel silently travelling the stars with its inhabitants never knowing where they are from.. unless John is somehow able to find out from the destroyed lab ..but seems unlikely..i guess their last chance of knowing would be to interrogate the Mr Hand but he was dying..) share