What are the elements that make Beowulf one the most celebrated epics of all time?
Beowulf's popularity and value are probably due to several factors.
For one thing, it is the only work of its kind to survive from its time period. Although we are all familiar with the much older Greek classics like The Odyssey and the Oedipus plays, there is nothing like that in ancient England. The English culture had simply not yet developed to that degree. In fact, the people of the island weren't even called “English” yet—they were Celts or Britons.
It is not even likely that the Beowulf story was created by an English poet—the story doesn't take place in England and its characters are Swedes (specifically Geats) and Danes. Most likely, the story was picked up orally by English monks and transcribed in monasteries, where the monks added some English and Christian elements.
Even so, Beowulf has value in its depiction of the warrior clan culture that dominated English life in the Anglo-Saxon era. The traits of this era are codified in the characteristics of the epic hero, of which Beowulf himself is a famous example. The epic hero typically embodies the most admired traits of its culture. Here we have what is probably the most important aspect of the Beowulf story—his loyalty, military skill, braggadocio, and valor are all valued traits of the Anglo-Saxon warrior. Beowulf, over the course of the story, gives us a glimpse, and one of the only existing glimpses, into that culture.
In many ways, the character of Beowulf is unlike most of his modern readers—he believes in fate, willingly accepts death, and brags endlessly about his own exploits. Nevertheless, he is the ancient superhero, his era's equivalent of our Batman, Spider-Man, and Superman.
So Beowulf is celebrated and studied because of its rarity—there isn't anything else like it—and because of the unique picture it provides of a time period that is known as the Dark Ages for good reason: we don't know much about it apart from works like Beowulf, and probably never will.