Two sets of people are responsible for their deaths: their families, and the lovers themselves.
If their families had not been feuding, there would have been no real barrier to them marrying (except, perhaps, their ages, and in the play it is noted that woman younger are married). The inability to resolve this conflict created the barrier.
Then, of course, Romeo and Juliet are responsible. They leap to conclusions and kill themselves.
fate duh... read the chorus