Is the play Wiliam Shakespeare's Romeo and Juliet based on a true story?

Shakespeare's play was based on The Tragicall Historye of Romeus and Juliet by Arthur Brooke (this is certain), which was based on a novella by Matteo Bandello. He got his material from Pierre Boaistuau and François de Belleforest, who got their material from miscellaneous French and Italian sources (all those were less certain).

So, basically, yes, but in such an indirect way I don't think it's relevant.

No. It's based on the myth of Pyramus and Thisbe. That's Ovid. He had other influences like the previous reply listed, but they were NOT true stories.