Is it possible for me to stay in Berlin for three months on $5000 not including living expenses?

Yes... if you are not paying for a hotel room. However if hotel comes into play that will be expensive.

I stayed in Berlin for the same amount of time for less than that, although I had 5,000 total to spend. Ended up spending about half that. A lot of it boils down to money management and smart shopping. There’s a great website called www.nomadnightlife.com that posts nightlife events, pub crawls, city cards (which are a God-send), food/drink related activities, and recreational stuff - all at discounted prices for Berlin and a handful of other major European cities.

Best of luck with your adventures!