According to poets.org, an epic poem is:

a long, often book-length, narrative in verse form that retells the heroic journey of a single person or a group of persons. Elements that typically distinguish epics include superhuman deeds, fabulous adventures, highly stylized language, and a blending of lyrical and dramatic traditions.

Beowulf qualifies as an epic poem on several of these criteria. First, it is book-length, and it is a story that is written in the verse format. It tells the tale of the main character, Beowulf’s, heroic journey to slay the monster and then much later kill the dragon. During the poem, he has several adventures and accomplishes nearly superhuman deeds, as when he kills Grendel.

Grendel is a monster that lives at the bottom of the swamp. One night, Grendel kills some of King Hrothgar’s soldiers while they are asleep. This leads the King’s people to fear Grendel, as the monster terrorizes them over a period of more than a decade. Not surprisingly, King Hrothgar wants to find a way to rid himself and his kingdom of the monster forever to escape Grendel’s monstrous, violent fury.

Beowulf learns about Grendel and leaves his home in Sweden to help King Hrothgar protect his kingdom and get rid of the monster. Beowulf has had many prior victories that make him the ideal person to tackle Grendel. Ultimately, Beowulf is victorious and kills the monster, who returns to the swamp to die.

This is not the end of the story, however. Despairing over the death of her son, Grendel’s mother wants to avenge his death, and Beowulf must do battle once again. Although there is peace for a period—Beowulf even ascends to the throne upon the death of the king—it is short-lived. Ultimately, Beowulf must do battle again against the dragon.

In short, reflecting its epic journeys and epic battles, plus the heroic actions of the protagonist Beowulf, the poem qualifies as an epic one. Moreover, as Beowulf proves himself to be a hero who voyages over a long journey to kill Grendel against odds that he will succeed, it is a story about an epic journey and epic undertaking.

Epics focus on the achievements of a particular individual. In Beowulf, the reader or listener is given an opportunity to follow the events and heroic achievements of the protagonist in the poem. The poem revolves around the battles fought by Beowulf until his death.

Exaggeration is an important feature of epics because it highlights the supernatural abilities or prowess of the hero. Beowulf’s strength is supernatural, and he is able to defeat both Grendel and his mother with his bare hands. He is also able to dive and stay underwater for hours without any form of technological assistance.

The setting is also vast, as seen when Beowulf travels to fight Grendel and challenges Grendel’s mother in some underwater underworld full of other monsters.

Beowulf can be defined as a blend of myth and heroic legend, which are features of an epic poem.

Beowulf can be considered an epic poem for a number of reasons, some of them outlined in the answer already given. Here are some additional reasons for considering Beowulf an epic:

• Epics often focus on single heroic figures (in this case, Beowulf).
• Epics often deal with war or with one-on-one combat, as in the three battles featured in this poem.
• Epics often involve heroic quests, as in Beowulf’s journey to Denmark.
• The hero of an epic poem often embodies the highest values of his culture.
• An epic poem is often “encyclopedic,” in the sense that the poem reflects an enormous number of different aspects of the culture from which it springs.
• An epic poem often contains, within itself, other genres of poetry, such as the lyric.
• Epic is often closely related to history as a genre.
• Epics often arise out of oral traditions in poetry, as is certainly the case with Beowulf.
• Epics often involve interactions between heroes and gods. In Beowulf, the Christian God is not immediately present in the poem but is often mentioned and discussed.
• Epic poems are often lofty in their styles of language, as is certainly the case with Beowulf.
• Epic poems often use so-called “epic formulas” – that is, phrases that are repeated and that are used in a variety of contexts. 
• Epic often features a certain amount of boasting, as when Beowulf is provoked, by Unferth, into boasting about his exploits as a youth.
• Epic poems often feature tragic deaths, as in the tragic death of Beowulf himself.

For a fuller account of some of these features of epic, see the appropriate article in The Princeton Encyclopedia of Poetry and Poetics (Princeton: Princeton University Press, 1993.

When one discusses what makes Beowulf an epic, one is engaging in analysis of what literary critics call genre, a way of classifying literary works by similarities in style, theme, and other features. Traditional epic is a genre with roots in oral culture. The earliest epics (Gilgamesh, Iliad, Beowulf, etc.) are often considered paradigmatic of the genre. Typically, epics are long narrative poems recounting warlike and heroic actions, with the leading characters being kings, princes, or other important people. The emphasis of epic is normally on action rather than psychology. There is usually some sort of religious or divine or supernatural element in  epics. Thus Beowulf displays many of the features typical of the epic genre, and is referred to as an epic.

How is Beowulf an epic?

The best way to answer this question is by looking at other epics, such as the Iliad and the Odyssey. If we do this, there are several features that make Beowulf an epic. 

First, Beowulf, like the Greek epics, deals with a hero and a voyage. Like Achilles and Odysseus, Beowulf is on a journey. Beowulf goes to the land of Danes to fight the monster, Grendel, and later he fights the dragon. 

Second, this might sound odd, but many epics start off as oral poems.  Beowulf shares this in common with the Iliad and Odyssey, which come from oral tradition as canonized by Homer. Related to this point, Beowulf, like the other epics, is written in poetry. Homer's words are written in dactylic hexameter and Beowulf in alliterative verse or alliterative poetry.

Third, there is a major contest in view, which is usually for glory. Beowulf fights two foes - Grendel and the dragon. Achilles fights Hector, and Odysseus fights the suitors. 

In light of these point, Beowulf certainly qualifies as an epic. 

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.