How many square miles did the city of jerusalem cover?

Depends on when you are talking about. This sounds like a Biblical Question so I'll answer it that way. 

Very surprisingly, during David's time the population wasn't much more than a couple of thousand according to archaeologists. The area covered was no more than 40 acres even during Solomon's time when Jerusalem was a very important city.  It may have had a larger population than in David's time, but the size was not much different.  

Forty acres was not much. To give you an idea just how big that might be, consider that a section of land is 1 square mile. 1 section = 640 acres.
40 acres is 1/16 of a square mile. That's like 5 city blocks in any direction and that might be overestimating in. Imagine that? I was very surprised to read that. The temple was massive when you read about it and so were the stables for the horses. 

Anyway, if I find out any differently, I'll post in comments. These are secular accounts and not Biblical ones, but I don't hear anyone seriously disputing this.