Beowulf embodies several traits that make him a textbook epic hero. Some of these traits include his larger-than-life physical strength, his courage in the face of extreme danger, and his loyalty to both his superiors and the men who serve him.

An epic hero is not perfect or immortal, but he is a sort of superhuman. He is larger-than-life in the sense that his qualities exceed those of regular humans. For example, an epic hero will have extraordinary strength. Beowulf displays this best when he fights Grendel and rips his arm from his body. Grendel is an imposing beast who has already easily destroyed a number of men, including warriors who are strong and experienced fighters. Beowulf's feat in killing Grendel perfectly displays how his physical strength sets him apart from others.

Beowulf is also incredibly brave and willing to put himself into physical danger to save the lives of others. First, he decides to go to Hrothgar's kingdom of his own volition despite hearing the horrific reports about the damage Grendel has caused. He seeks out Grendel's mother in her cave, so he is putting himself into a foreign environment to face off with an angry, violent monster. Later in his life, when he is much older, Beowulf sacrifices himself fighting a dragon to protect his kingdom.

Finally, Beowulf is loyal to his superiors and to the men who serve him. Before he goes into a battle or fights any of the monsters he faces, he makes sure his men will be taken care of. He is their lord, and they swear fealty to him. He is responsible for them, and he holds up his end of the bargain. Beowulf is repaid by one of his men, who finishes off the dragon after it has mortally wounded Beowulf himself. His men also honor him after his death, memorializing him as the epic hero he has proven himself to be.

Beowulf is an epic hero for many reasons. Three of Beowulf's most epic characteristics are strength, honor, and the ability to rule justly. 

1) Strength: First and foremost, Beowulf is strong, and is known to possess power greater than any normal warrior. This is evidenced by his ability to wrench off the gigantic Grendel's arm and by his ability to contend with Grendel's equally monstrous mother. In short, Beowulf, like Hercules and Achilles, is far stronger than conventional warriors, and his superhuman abilities elevate him to an epic status.

2) Honor: Beowulf is also described as being honorable and virtuous. This characteristic comes into play many times, but it is most evident in Beowulf's commitment to fighting Grendel in hand-to-hand combat, rather than using man-made weapons. Beowulf reasons that Grendel does not use weapons, and so he will likewise eschew any armament besides his own strength. Thus, Beowulf shows that he follows a respectable code of honor, extending respect even to monstrous foes.

3) The ability to rule justly: Beowulf rounds out his epic qualities by proving himself to be a capable ruler. He clearly does not want power, as he initially refuses the throne of his country and only becomes king when it's clear there is no other option. He then goes on to rule for fifty years, and it seems as if the period of his rule is marked by peace and prosperity. As such, Beowulf is epic not only because he is a skilled fighter, but also because he is capable of justly ruling the kingdom entrusted to him.

There are four characteristics that all epic heroes must have—they are Glorified, On a Quest, Ethical, and Strong. Beowulf is glorified not only because of his noble birth, but because of his past deeds and accomplishments. Beowulf is on a quest, not only at the beginning of the poem (to kill Grendel), but throughout the poem—his mission is to gain more fame and fortune. He is considered ethical because of his protective nature and willingness to help the Danes with their problem. Although it is debated, I would argue that Beowulf gradually loses this trait through the story. Lastly, Beowulf is strong. His strength is exemplified through his ability to complete tasks no other man can complete, such as ripping off Grendel's arm, lifting and wielding the giant-forged sword, and carrying Grendel's head without assistance.

An epic hero has to perform heroic deeds. Beowulf kills Grendel, Grendel's mother, and the dragon, all of which are heroic deeds that no one else was able or willing to take on. The foes that Beowulf fought were all beyond human strength and capabilities and required superhuman abilities to bring down. Beowulf performed these deeds for the good of others, which is another quality of an epic hero. In the end, though, it was the human mortality of Beowulf which brought him down when the poison wound caused by the dragon resulted in Beowulf's death.

An epic hero is, in short, the hero of an epic.  An epic is a story, usually a poem, where a character goes on a quest.  There are some characteristics that an epic hero often has.  You can read about them in my first link.  Here are some ways Beowulf is an epic hero. 

(Please note that there are different versions of the poem.  My quotes come from the enotes PDF.  See my second link.)

Beowulf is a strong leader.

Beowulf is very persuasive and charismatic.  He leads his men and they are honored to follow him.  Sometimes the mark of a true epic hero is someone who can lead others and get them to go along with his will, and even follow him into danger.

Now many of Beowulf's band brandished ancestral blades, wanting to save the life of their leader, the proud prince, if such they could do. (ch 12,enotes pdf,  p. 18)

Beowulf is clearly loved by his men, and they will follow him even into great danger.  Beowulf provokes bravery in his men.

Beowulf has amazing physical prowess.

An epic hero is usually strong, often unusually so.  Beowulf is strong enough to battle monsters with his bare hands, without a sword. 

The monster desired to fling himself free, if at all he could, and fly far away to the fens—he knew that his fingers' power was in the grip of a fearsome foe; this was a dire march to Heorot that this devastating beast had made! (ch 11, p. 17)

Grendel doesn’t stand a chance!  Beowulf’s strength leads to even more bravery, and makes others admire or fear him, depending on what side they are on!

His exploits are sung of and heralded.

Every good hero has to have a cheering section!  All of Beowulf’s triumphs are sung, especially the defeat of Grendel.

Then gray-haired clansmen, many youths, and stalwart warriors rode back in high spirits on horses from the mere, and Beowulf's victory was recounted. Many a man said that among all the seas of the world, south or north, that none of the other shield-bearing warriors under the expanse of heaven's vault were more valiant or more worthy to rule! (ch 13, p. 19)

By recounting Beowulf’s glory over and over, he becomes a folk hero to all of the people.  They admire him and love him for saving them. 

The hero also gets gifts, of course.  Often these are useful to him later.

Then did the son of Healfdene present to Beowulf a banner woven of gold as an ensign for the victory, an embroidered flag of battle, a helmet and a coat of mail, and a precious sword that was seen by many when they brought it before the hero. (ch 15, p. 21)

Beowulf gets a heroic sword to fight Gredel’s mother with.

The epic hero “dies” and is resurrected.

A significant part of the hero’s journey is where the hero is reborn.  Beowulf kills Grendel, but then has to face and defeat Grendel’s mother.  He does so by going deep under the dark waters, a symbolic death.

The life of the son of Ecgtheow, prince of the Geats, would have ended there underneath the wide earth if his armor of war, hard net of battle, had not aided him; and the Holy God, wisest Maker, wielded the victory. The heavenly Ruler championed his cause, and he soon stood on his feet again. (ch 22, p. 28)

So you can see that the epic hero is more than the sum of his parts.  He is an inspiration to others.