I think that Istanbul has as good a claim to be an Eternal City as Rome has - perhaps even better, since Istanbul has been an important city continuously since Constantine chose it has the capital of the Roman Empire in the 4th century, thus driving Rome into decline for centuries. One really visible sign of this continuity is Aya Sofya, the huge Byzantine church built in the 6th century which was preserved virtually intact over the ages because when Moslems captured the city they converted it to a mosque. No building in Rome from that time survives (with one exception) because they were all used as stone quarries by mediaeval peasants. (The exception is the Pantheon which survived because it was made of non-reusable concrete!)
I walked around the old core of the city quite a bit today. No grand boulevards like in Paris, which gives the walker the upper hand. The only real hazard is the plethora of relentless carpet salesmen! I guess I'm too polite, but I've been dragged into three shops "while I get my business card" - funny how they never have a card on them. So far I haven't bought my carpet yet (as they like to put it), but I did succumb to a bookmark with a verse from the Quran written in beautiful calligraphy.