We actually know that both Juliet and Romeo come from noble families, and therefore would not have, or even need, any occupations. We are given a few clues about this, but one of the best clues is found in the play's Prologue in which the two families are described as "alike in dignity" (1). The term "dignity" can be translated as referring to "elevated rank" or "station," thereby proving that both families have a high social station (Random House Dictionary). The second clue is that both characters' mothers are addressed as Lady Capulet and Lady Montague respectively. A Lady is the official title of the wife of a Lord, hence we know that both Capulet and Montague are Lords.
Since both families are of nobility none of their family members would actually have occupations. Their fathers own land from which they receive a substantial income, therefore, even their fathers do not have occupations, but rather live the leisurely lives of noble gentleman.
Juliet, as a woman, would have been refused to have an occupation in those days. However, if she had married County Paris, she would have married a man who's noble rank is far higher than her father's and would have become a Countess. She was also expected to inherit her father's estate. At the age of 12, Juliet is too young to inherit the estate, but we know that her father has declared her to be his heiress because we see him threaten to take it away from her if she refuses to marry Paris, as we see in the lines:
Hang, beg, starve, die in the streets,
For by my soul, I'll ne'er acknowledge thee,
Nor what is mine shall never do thee good. (201-203)
We can translate the line "what is mine" to refer to all that Capulet owns, including what would be her inheritance.
Likewise, Romeo will also inherit both his father's title and his land. Had Romeo and Juliet lived, Juliet would have also become a Lady, like her mother.
Romeo and Juliet do not have occupations, that we know of. They are both very young (Juliet is 13, and presumably Romeo is as well). There is no clue as to if (or for that matter what) their occupations would or could be. They are simply teenagers, jobless. If it was a detail worth noting, Shakespeare would have included the information, but knowing if, or what, Romeo and Juliet do for a living is irrelevant to the overall plot line of the play.