Sarajevo was the headquarters of First Corps of Army of Bosnia and Herzegovina, which in 1992 had 34 500 soldiers in Sarajevo, and 40 500 at the end of the war. The troops lacked heavy weapons so they couldn't break out of the city, but they could still defend it heavily. Especially due to atrocities commited early in the war in Eastern Bosnia, which made a lot of soldiers fight to the death, or commit suicide rather than be captured. As you can imagine fighting such a determined resistance is very hard.

Serbs on the other hand lacked the manpower to commit to a full assault on a city of half a million people, they had around 20 000 men. They had a lot of heavy weaponry and tanks, but lacked manpower, thus they commited to siege, rather than assault.

That is not to mean that Serbs didn't try. Early in the war Serb special forces tried to infiltrate the city and capture key locations within the city, but they were defeated with a lot of dead.

They also managed to capture the president of Bosnia and his daughter, but due to quick thinking by Vice president he was rescued.

To add to what /u/Hrulj said: urban fighting is probably the most difficult and costly kind of ground fighting today. If you're an attacking force, you need large amounts of manpower, firepower, or both. Even then, it's going to be an affair that will result in more casualties.

Simply put, even for a first-rate power like the United States, capturing a city on the scale of Sarajevo (both in size and population) would be an enormously difficult and costly task. The Serbians did not have the capability to take it in the face of the defenses that it had.