Juliet is in a dilemma because she is already married to Romeo, who has been banished for killing Tybalt and her parents have no idea and have arranged for her to marry Paris. Knowing how ill-tempered her father is, she can't tell him she is married to Romeo, their enemy, so she must come up with another plan to get out of marrying Paris. This is when she turns to Friar Laurence to help her.
Juliet's core dilemma is the essential plot of the play. In other words, her dilemma is that she falls in love with a Montague. That family and the Capulets are enemies of long standing.
Add to that the fact that another man is pressing her father to marry her, and you've got family loyalty vs. love, twice over.
Since her father is a powerful man, if she were to run away, he would have men from all around to capture her and possibly kill her. However, she can't stay because if she does, she has to marry Paris (or risk disownment) but she can't marry Paris because she's already married to Romeo.