How often have we heard--even in rather pedestrian movies--a person or character remark, "You know, So-in-so is a real Renaissance man." The "intense moral feud" is as kc4u so eloquently has described, indeed, a universal and existential state. This is the very feud that Saul Bellows's Herzog finds himself mired in, for one example.
Whether or not the modern world can relate to the Renaissance, we are surely attracted to it. We love the big dresses and the elaborate art and the "grand"-ness of it all. The music still shows up in our world (think Manheim Steamroller), as does the art and even the clothing (although I doubt we'll ever see the pantaloon style and hose for men come back into style anytime soon!). We're fascinated by the characters of the Renaissance, particularly the Medicis and the Tudors. The word falls trippingly off the tongue and we are consistently looking for the next "new" thing, often by looking back at what was.
In terms of the literature, this is the stuff we still read and still enjoy. The characters of Renaissance literature are alluded to and replicated in both modern literature and in movies. These classics won't be phased out of our world anytime soon.
This is the sort of question that is really one that each individual must answer. However, for the sake of generalization, I would have to say that of course we can relate to literature from the Renaissance. In support of this point of view, I would offer a novel, Don Quixote, and basically the entire canon of Shakespeare's plays. We can relate to these texts, even in the 21st century, because their characters are ones that move and touch our understanding of what it means to be human, no matter what time period we live in.
Don Quixote is understood by most scholars to be one of the very first (if not first) modern novels. As Unnamuno (from the Enotes study guide of the novel) states below, the book continues to affect and move readers:
Ever since Don Quixote appeared in print and was placed at the disposition of anyone who would take it in hand and read it, the book has no longer belonged to Cervantes, but to all who read it and feel it. Cervantes extracted Don Quixote from the soul of his people and from the soul of all humanity, and in his immortal book he returned him to his people and all humanity. Since then, Don Quixote and Sancho have continued to live in the souls of the readers of Cervantes' book and even in the souls of those who have never read it.
And I could include many, many quotes about how readers and playgoers throughout history have been moved by the works of Shakespeare. The sheer number of theatre companies which consider themselves to be Shakespeare Festivals throughout the world attest to the continuing attraction and ability to relate to the characters of these great plays. Here's a quote from the Entoes introduction to Shakespeare the Dramatist:
Few dramatists can lay claim to the universal reputation achieved by William Shakespeare. His plays have been translated into many languages and performed on amateur and professional stages throughout the world. Radio, television, and film versions of the plays in English, German, Russian, French, and Japanese have been heard and seen by millions of people. The plays have been revived and reworked by many prominent producers and playwrights, and they have directly influenced the work of others.
Please follow the links below for more on Don Quixote and Shakespeare.
In my view, the modern world was born in the womb of the European Renaissance. The Renaissance philosophers, thinkers, artists, authors and commentators did really set the ball rolling. How can we delink ourselves with the Copernican Revolution, with Thomas More's Utopia, Machiavelli's The Prince, with the plays of Shakespeare and Marlowe, with Cervantes's Don Quixote, with the Metaphysical poets like Donne and Marvel?
In fact literature and culture must always be a matter of looking and relating back in time. All great works of man are at once born out of their milieu and moment and universal. Is the self-divided Prufrock in T.S.Eliot's Love Song not related to Shakespeare's procrastinating Hamlet? Is the intense moral feud in the soul of ambitious Macbeth just a simple figment of the by-gone years? Think about Marlowe's trouble-torn Faustus. Consider how the ancient king Pericles, appearing in Shakespeare, reappears in Eliot's poem Marina.
Renaissance marked an end of the medieval geocentric/ theocentric universe and ushered in the new heliocentric/ethnocentric universe. That was the beginning of the modern world. We have to relate our thought to the Renaissance. It is indeed mandatory.