"I have heard moreover that the monster scorns in his reckless way to use weapons; therefore, to heighten Hygelac's fame and gladden his heart, I hereby renounce sword and the shelter of the broad shield, the heavy war-board: hand-to-hand is how it will be, a life-and-death fight with the fiend." (433-440)
Beowulf makes his battle with Grendel more than a simple slay-the-monster task. By announcing that it will be a hand-to-hand combat, he gains extra glory for himself and the Geatish king, Hygelac, turning the contest into a feat of strength as well as a fight against evil.
Beowulf, Ecgtheow's son, replied: "Well, friend Unferth, you have had your say about Breca and me. But it was mostly beer that was doing the talking. The truth is this: when the going was heavy in those high waves, I was the strongest swimmer of all." (529-534)
It's important for Beowulf to show strength even when there isn't an important heroic task to be accomplished. When there aren't demons or dragons to fight, he gets into these, er, "swimming contests" with other warriors.
The monster's whole body was in pain, a tremendous wound appeared on his shoulder. Sinews split and the bone-lappings burst. Beowulf was granted the glory of winning; Grendel was driven under the fen-banks, fatally hurt, to his desolate lair. (814-820)
How strong does a medieval Scandinavian hero have to be? Strong enough to rip a demon's arm out of its socket. That's how strong.
The captain of evil discovered himself in a handgrip harder than anything he had ever encountered in any man on the face of the earth. Every bone in his body quailed and recoiled, but he could not escape. He was desperate to flee to his den and hide with the devil's litter, for in all his days he had never been clamped or cornered like this. (749-756)
Beowulf's first great exploit during the epic (we're not counting his swimming contest with Breca, which we only see in flashback) is a combination of strength and skill: incredible strength, with which he clamps down on Grendel's arm, and skill in wrestling, with which he finds a joint-lock in which to hold the demon.
The story goes that as the pair struggled, mead-benches were smashed and sprung off the floor, gold fittings and all. Before then, no Shielding elder would believe there was any power or person on earth capable of wrecking their horn-rigged hall unless the burning embrace of a fire engulf it in flame. (774-781)
Beowulf's strength is depicted as nearly, or maybe actually, superhuman. His wrestling contest with Grendel causes more destruction than anyone had thought humanly possible.
"Be acclaimed for strength, for kindly guidance to these two boys, and your bounty will be sure." (1219-1220)
In this passage, Queen Wealhtheow praises Beowulf for defeating Grendel and asks him to remember the rights of her sons in the Danish kingdom. It's particularly interesting that Wealhtheow equates "strength" with "kindly guidance" – apparently being a strong man and being a wise one are pretty closely associated in this culture.
There was less tampering and big talk then from Unferth the boaster, less of his blather as the hall-thanes eyed the awful proof of the hero's prowess, the splayed hand up under the eaves. Every nail, claw-scale and spur, every spike and welt on the hand of that heathen brute was like barbed steel. Everybody said there was no honed iron hard enough to pierce him through, no time-proofed blade that could cut his brutal, blood-caked claw. (979-989)
It's not enough for Beowulf to tell stories of his physical prowess; he also needs to display concrete evidence of his deeds. For example, Exhibit A: the severed arm of Grendel hanging from the rafters of Heorot Hall.
She came to Heorot. There, inside the hall, Danes lay asleep, earls who would soon endure a great reversal, once Grendel's mother attacked and entered. Her onslaught was less only by as much as an amazon warrior's strength is less than an armed man's when the hefted sword, its hammered edge and gleaming blade slathered in blood, razes the sturdy boar-ridge off a helmet. (1279-1287)
Although the usual pattern of a three act plot suggests that Beowulf's second battle should be more difficult than his first, Grendel's mother is actually a little bit weaker than Grendel himself. Still, the narrator reminds us that she's a vicious, violent, unbelievably strong opponent. She's compared to an Amazon, a member of a mythical tribe of female warriors in Greek mythology. Fighting Grendel's mother is like fighting an Amazon, whereas fighting Grendel is like fighting a male warrior. What's the difference? Well, the Amazon might be slightly less strong because she doesn't have the same physical build as a man – but when a bloodthirsty warrior slices the crest off your helmet with a sword, you don't really stop to think about the warrior's gender, do you?
"I would rather not use a weapon if I knew another way to grapple with the dragon and make good my boast as I did against Grendel in days gone by. But I shall be meeting molten venom in the fire he breathes, so I go forth in mail-shirt and a shield." (2518-2524)
Beowulf is careful to explain why he allows himself the advantage of armor and weapons in his battle with the dragon: it's because the dragon has its own special advantages: poisonous venom and the ability to breathe fire. It's not enough for Beowulf to battle the dragon. He has to emphasize that, in doing so, he really is meeting the creature on a level playing field, demonstrating his own strength and prowess, not just using better weapons.
Inspired again by the thought of glory, the war-king threw his whole strength behind a sword-stroke and connected with the skull. And Naegling snapped. Beowulf's ancient iron-grey sword let him down in the fight. It was never his fortune to be helped in combat by the cutting edge of weapons made in iron. When he wielded a sword, no matter how blooded and hard-edged the blade his hand was too strong, the stroke he dealt (I have heard) would ruin it. (2677-2687)
Throughout Beowulf, swords snap, melt, and otherwise fail their owners. During Beowulf's final battle with the dragon, the narrator explains that our hero is just too strong for the blades of the swords forged by men. It's just one more hint that Beowulf's strength is more than human, mythic in its proportions.