Shakespeare's Romeo and Juliet is not based on a real story, but it is not original to Shakespeare either. An important source is the Roman writer Ovid's Metamorphosis. One of the stories in Ovid's work is Pyramus and Thisbe, about two Babylonian lovers. Their love is forbidden due to a family feud, and both the lovers end up committing suicide over a mix-up. Much closer in time was a sixteenth century poem called The Tragicall Historye of Romeus and Iuliet and William Painter's The Palace of Pleasure, a collection of stories including one about Romeo and Juliet.
While the play is based on a fictional tale, it is plausible that there is some factual foundation for the story. At a time when families exercised much more control over marriages, it is likely that two young people in love would have been thwarted by two families in conflict with each other. But we don't know of such a story, only of fictive accounts.