The exact size or population of the city are unknown. Mr. Snippy once mentioned 3 billion people surfing the internet via ANNET at all the time but it is possible he refered to humanity as a whole and not just the citizens of Eureka. Pilot speaks of 1 million voices of Eureka in his blogs.
When the ANNET went berserk, it reprogrammed all citizens connected to her to serve the machines, basically turning them into zombies.
Eureka was finally destroyed, when the last human survivors started a nuclear counterattack on the ANNET.