An epic story always has some sort of hero figure. There is also usually a grand battle of some sort. That would be why I consider The Iliad to be an epic, but not the Divine Comedy. I guess Harry Potter could also be an epic.
I disagree to an extent. Gilgamesh, for example, is certainly an epic. It does not centre around a battle. Likewise, the Atrahasis is a flood myth, I think it has characteristics of an epic. While, for a number of reasons, I'm not quite willing to describe the Divine Comedy exclusively as epic I think it is much closer to being one than is Harry Potter. At the very least with the inclusion of Virgil, Dante closley aligns his work with epic tradition. To the OP: read The Hero with a Thousand Faces by Joseph Campbell. Beyond the style in which they are composed, I think heroism, the intersection of man and the divine, larger than life events, and the journey are common characteristics of epic works.
Ok perhaps the grand battle part isn't true, but the part about having an epic hero is true. Harry Potter is more of an epic hero than Dante is, IMO. I think the main reason The Divine Comedy is considered an epic is for its length. The hero doesn't really go through any trials nor does he perform heroic deeds.
He may not physically battle anything, but in facing the layers of the Inferno and circles of Purgatory, he is forced to confront his own deficiency, understand the nature of sin and the justness of the punishments inflicted on the sinners and the atonement of the penitent. He separates from his community, undergoes a journey into the wilderness, descends to the underworld, confronts the divine, and gains a boon. While he may not be much of a 'hero' in the sense we normally think of one, I think it can be argued he goes through several stages of the heroic journey.
I think of it as more of a journey to gain knowledge. That does not automatically make him a hero. Otherwise, many other scholarly characters in literature could also be considered heroes for gaining knowledge. I guess we differ on the definition of what a hero is in literature.
I do think there's more to it than a quest for knowledge, but I also think it's a matter of how you read it. I read a lot philosophy and theology into the Divine Comedy and so it is probably not surprising I find types from mythology in it. This discussion makes me want to get my copy off the shelf and read it again, it really is an amazing and deep work.
An epic seems to concern the lives or existences of many things, usually. That's why wars and battles are usually in epics, because they concern life and death for hundreds and thousands of peoples, entire kingdoms, you get the idea. Then again, some epics like Dante's Inferno or the aforementioned Epic of Gilgamesh concern only one person again... SO I could say, I guess, that epics concern larger-than-life kind of things. Dante's Inferno concerns a guy going down into hell, which is obviously not your everyday normal thing. The Illiad concerns a bunch of immortals, half-immortals, and mortals battling out their guts, the Odyssey about a warrior's long and many-danger filled journey home. Not your average thing.
The epic concerns itself with a larger-than-life hero who accomplishes amazing, impossible tasks that pertain to something crucial to his life, a large setting with some sort of supernatural force(s) in it, and the hero must have a combination of flaws and positive traits that make him seem human despite the quest. Classical examples are the Odyssey and Inferno. A modern example is Harry Potter. Another modern example is Star Wars. The work's quality has little to do with whether or not it should be classified as an epic.