As someone who has successfully dated women "out of my league," it would be silly of me to say such a thing is impossible.
That said, I would urge you to take a long and hard look at the men she has been known to have dated: Is there something about her personality you genuinely admire? Are you looking for bragging rights that you went out with her once? Is there some aspect of her character that you admire and desire for her to be your life-partner/girlfriend/spouse?
Aside from the obvious reason of her beauty...
...if your reason for fixating on JLH is just her looks, fame, or on-screen persona, you are going to suffer the same fate as all her other squealing fanboys who have unrequited sexual fantasies about her; they have no grasp of her as a real person, complete with real human flaws ... like EVERY OTHER WOMAN on the planet.
Would you still desire her if you discovered that JLH farts a lot in bed, has horrid morning breath, or picks her boogers and wipes them on the bathroom counter?
Still determined to go out with JLH for non-shallow reasons?
Remember to bring your A-Game because some of the best-looking, most-talented, wealthiest, most charismatic men on the planet (see the list above of her former paramours) have courted, seduced, kissed, and made love to JLH.
You don't have to be a movie star, but you have to bring SOMETHING extraordinary to the table for her to even consider dating you.
An impressive but non-Hollywood skill/profession/hobby would work (and, in this regard, be more effective than if you were yet another handsome millionaire actor - she's clearly had ample opportunities to be with many, and married NONE of them).
I'll make the optimistic assumption you're a world-class surgeon, Olympic-caliber athlete, US Navy SEAL squad leader, concert pianist, PhD scientist working on a cutting-edge NIH-funded project, a GrandMaster-League StarCraft II player, or something similarly awesome.
You're not in the Hollywood "scene," and you don't want to approach her as yet another a fan seeking her romantic attentions - she already has a stalker list a mile long, and adding your name to the bottom of her list of restraining orders does you no favors.
Here's what you do.
Find out what charities she actively supports and make a generous donation with no strings attached (don't be the creep that says "well, I'll write a check if I get to meet her").
Believe me, an out-of-the-blue major contribution to a nonprofit from a stranger will definitely get her attention. The Executive Director WILL approach you, asking what motivated you to write this $10,000 (or whatever) check to their nonprofit/foundation. Let this person know that you were first alerted to their cause by their spokesperson JLH and were very inspired by her endorsement.
You admire the work they do and wanted to see them fulfill their mission doing [whatever].
If the Executive Director is smart, s/he will ask if there's anything they can do for you, at which point, you say (as if it just occurred to you) "Hey, I know she's a huge celebrity, and I'm totally okay if the answer is 'no,' but I would love to catch her for lunch next time she's in town to do a promotional thing for your foundation/nonprofit, if that's something you can pull off."
And there's your in.
If you manage to get this far - message me, and I can walk you through how to initiate and maintain a romantic relationship with an A/B-list celebrity for [X] months as a commoner of modest income/net worth (that is the limit of my expertise).
The rest is up to you ...More questions on Dating and Relationships: