Shakespeare never tells us why the families hate each other. It could be that the feud is so longstanding that the families themselves don't know why it started. The Montagues have always hated the Capulets and vice versa.
Some scholars see similarities between Romeo and Juliet and the story of Pyramus and Thisbe, who were also kept apart by feuding parents. Shakespeare uses the story of these lovers as the play being rehearsed by Bottom and his crew in A Midsummer Night's Dream.
Verona, Italy, where the play is set, was under the rule of Venice during Shakespeare's time. The city was known for its artists and architects; indeed, Verona has been called a "social, cultural, and economic fortress." Perhaps Mr. Montague and Mr. Capulet were rival builders or rival painters.
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No one really knows. It is an age-old feud that no one has bothered to explain to either Romeo or Juliet, but it just understood that every member of both families and even down to the servants of those families participate in on a daily basis. Therefore, every time anyone from either household encounters a member from the opposing household in the streets, there is a quarrel of some kind. It is so bad that the Prince declares that the next person who causes disruption of the peace in his streets will be put to death.
This is the reason Romeo has to run away after killing Tybalt for his revenge for Mercutio's murder. If Romeo stays in the city, he will most certainly be put to death for his disturbance of the peace and bloodshed.
The feud has been in place so long, probably none of them could explain the true reason it began...sort of like the Hatfields and McCoys.
There is not a specific reason given.