Romeo and Juliet have risen from the grave in search of braiiiins! 
I’ve always loved this play, and I agree with the points Fretful Porpentine brought up–that this was an age when people believed true love happened at first sight (it may take suspension of disbelief, but if we can see the play through that lens it adds poignancy), that the quality of Romeo’s language when he talks about Rosaline and when he talks about Juliet is a clue to the depth of his feelings, and so on.
And the Asimov interpretation? I’ve read it myself, and although it’s interesting, I can’t ultimately buy into it. Asimov (based on that one scene at the party) concludes that the feud is a non-issue among anyone except servants and one or two malcontents, and wouldn’t be an issue except for Romeo and Juliet’s actions. Except that the prologue (supposedly an “objective” voice), describes “ancient grudge” and “new mutiny” the “fatal loins” of “foes”, not a dying-out feud.
What’s more, during the first brawl, Lords Capulet and Montague, supposedly the older, wiser heads of the family, don’t try to calm things down and talk sense into the young hotheads…they immediately try to get into it themselves, even though their wives try to stop them getting themselves killed. The Prince, a few minutes later, refers to several previous, recent brawls, “bred of an airy word/By thee, old Capulet and Montague.” Capulet later tells Paris that it’s not hard for old men like him to keep the peace, but it seems at odds with his earlier actions, and he might just be blowing smoke to make himself look good to his daughter’s suitor. (As we see later, when he commands Juliet’s marriage to Paris although he said earlier that he would take her wishes into account, he can be expected to say one thing and do another.)
Which leads us to that scene at the party where Capulet seems to have no problem with Romeo being under his roof. So why is that if the feud is so bad? Well, just a few hours ago the Prince told him “one more brawl and it’s your head,” and two kinsmen of the Prince are right there…kinsmen that can pass along word to the Prince that Capulet’s being a good boy and not making any trouble. (That may also help to explain why Capulet is so eager to marry Juliet to Paris once Mercutio, the Prince’s other kinsman, has been killed off in yet another brawl…anything to get back in the ruler’s good graces.)
I agree that it’s a little hard for modern audiences to suspend their disbelief, but if you can, you’re in for a treat. Still, there’s Beatrice and Benedick (from Much Ado) as an example that Shakespeare can do couples that are both romantic AND realistic. Beatrice and Benedick have known each other a long time (and it’s hinted that they had some prior involvement before Benedick got cold feet and backed off), know each other’s strengths and weaknesses, faults and virtues, and love each other deeply anyway.