I would point you to a real world example...North Korea. Currently, the country is isolated from the rest of the world partially due to its own choice and partially due to sanctions placed on it by the outside world. Some things to keep in mind would be the resources that such a city would require to maintain itself. If you are looking at today's world, that is a massive feat to undertake. Today's level of technology was achieved by specialization: many finding that which they excel at and focusing their energies there while trading the products of their labors for their other needs from others who specialized in other ways. If the city in question is to be of an equivalent level of advancement to the rest of the world, then it will have to compete by maintaining all of this specialization within itself. Another question that should be asked is how the city would have managed to have advanced in pace with the rest of the world. Without external factors pushing it (enemies/war), there wouldn't be the motivation to develop fast which should put it far behind most countries.
Food production is also a necessity. Farming requires land. Even the city-states of Ancient Greece had a lot of land to them with which to produce food. This will force your city to be a small landmass to be sustainable. Even if you find a way to produce food inside the city, the city must have developed from a lesser sophisticated one that required the land at one time.
Another question to ask is the amount of time that the people have been isolated. For one, they would have their own culture all their own (or a derivative of a modern day one with whom they split from in the distant past). If they have been isolated for a couple thousand years, they would still be considered human; however, they would have likely developed certain genetic mutations that wouldn't be present in the outside world that could alter their appearance and/augment them differently.
A last note is how aware are the citizens of the outside world? Going back to North Korea as an example, the upper echelon is aware and does participate in the outside world while the masses are sheltered within and controlled. How are citizens who want to expatriate treated/handled? Are they allowed to leave and under what circumstances? Are they imprisoned for their desire to leave? Your form of government will also be key as democracy likely wouldn't maintain secrecy too well with open thought.