Well, if you were expecting an "yes", I'm sorry to disappoint
The balcony is in fact real, the house belonged to the Bartolomeo family, Verona was ruled by Prince Bartolomeo I della Scala for real, ages ago though.
The story of Romeo and Juliet was inspired by other stories written before. All in all, Romeo and Juliet's story is not real, but is it based on real facts.
There's also a legend that says that a girl killed herself out of love because she and the man she loved couldn't be together.
The first one who wrote about this legend was actually Luigi da Porta and not Shakespeare. He wrote about "Giulietta" and "Romeus" and created the fictional characters of "Mercutio" and "Tybalt". Then, inspired by Luigi da Porta, Matteo Bandello wrote
Romeo and Juliet and so on
The balcony is physically real, but the story isn't, it's just a legend written and rewritten until it became famous.