Heroism often looks different based on who is viewing the hero and what the cultural values are. If we focus only on action, then many ancient heroes would fall short of our current standards. By this reasoning, we could fault Beowulf for being a warrior for hire when he helps Hrothgar, not exploring Grendel's motives, and then fighting the dragon alone and leaving the Geats undefended at the end of the epic.
At the same time, he comes to the aid of an aged king whose country was devastated by an invading aggressor, puts a quick end to a hostile terrorist, and sacrifices himself for his country.
The reason these stories remain classics is that they allow us to return to the same basic story and review our own values, debating exactly these questions and resisting a simple answer to the problem of how one should behave.
Beowulf is an exemplar of the Germanic value of comitatus, which was a code of honor based on loyalty, generosity and bravery. In this regard, the young warrior Beowulf lives by the code and exceeds all others. As a king, he might more properly have insisted a younger man fight the dragon, but clearly none were willing to do so. Only after Beowulf has shown the way and been injured does anyone step forward.
Today, we tend to value heroes who inspire others to act or who can use intelligence to avert bloodshed. We prefer a humbler hero who minimizes their personal glory, and who achieves results for a larger group of people. Heroes are valued for ideas and for rhetoric rather than brawn. In Anglo-Saxon times, the hero myth valued the singular warrior who took on monsters or insurmountable odds and showed a unique physical and militaristic strength.
The industrial revolution and the use of machines in battle reduced the type of glory one could expect on a battlefield, but in other ways we would still likely consider a person a hero if they are able to show a certain excellence in an area where others attempt but do not achieve fame. In many respects, were Beowulf alive today, his heroism would be most acknowledged in an athletic competition, where are great athletes are celebrated in many of the same hyperbolic terms used for Beowulf.