How many stars can stay close to each other without collapsing? Is it possible for two stars to exist close to each other? 

"Close" is relative; let's assume that two stars are close to each other if they are at the center of the same solar system. 

It's possible for this to happen with 3 stars. But what about more? How many stars can be the center of exactly one solar system without collapsing because of gravity?
Well, if they have an angular momentum with respect to their center of mass, or in other words , if they are in orbit, gravitational collapse is not the issue.

This is what happens when three bodies are in orbit around each other. Don't get disappointed so soon. There is indeed a way to avoid this. If the three bodies are of comparable mass, and one of the bodies is situated at L4 or L5, tuning the orbital parameters may work just fine, but then you gotta worry about tidal forces. 



So, if the size of our third star exceeds the area given to it at L4/L5, it will experience tidal forces and will start ejecting mass to either of the two other stars, or both.

But as Zolan commented above, astronomy with an infinite universe has a possibility for everything. What are listed above are the most common problems which arise, that is the reasons which are mainly responsible for making such systems highly unlikely of existence.

But here's a system which works the way you described : 

A binary system exists, and the third star is far enough that it experiences the gravity as that of a single star. And the nearest planet(as you want it to be a "solar system"), is so far that it experiences the gravity of the three stars as that of one. Now, since "close" is a relative term, for the planet, the three stars are appreciably close.

So, the answer to your question, YES and NO