What drastic measure does juliet take to avoid marrying Paris?
Juliet takes a potion that will make her look as if she were dead for long enough for her to be mourned and buried. The Friar's plan is to send word to Romeo about her fake death and the Friar and Romeo are to watch over her until she wakens.
I do not think Juliet really wants to die. She does say that she'd rather kill herself than marry Paris and have sex with him. Although she says this, she leaps at Friar Laurence's suggestion that she fake her death. This would indicate that she is desperate for a solution but is not completely without hope. Yet. She wants to live.
She does not take the potion lightly. In fact, prior to taking it she considers many things that could go wrong: the Friar could be giving her real poison to kill her and protect his reputation; she could wake too soon and suffocate in the stale air of the tomb; or, she could wake to the spirits of the dead, go crazy and dash her own brains out with an ancestor's dead limb.
Her desperation is evident in that despite all she considers, she remains willing to risk death and to put her body into a death-like state in the hope that she can find a way out of her situation. Of course the one solution she does not consider is to tell her parents the truth. She also does not consider the one thing that could destroy her plan completely--that is if Romeo were to somehow think that she were actually dead.
Juliet drinks a potion which causes her heart to almost stop, which results in her skin being cold to touch. The potion's effectiveness lasts for two days. By doing so, Juliet will be placed in a morgue, such that Romeo will visit Juliet in the morgue upon receiving Friar Laurence's letter.