In this context it means big, grand, of extraordinary proportions.

Hey O ~ The word "epic" is used now as a sort of a 'high water mark," for anything in a series. So, we all sulk, (get blue, depressed, negative), but this sulk is the biggest, the best/worst, the most massive sulk on record for this particular person in his/her life, to date / so far. That's what makes it epic. There may be a more epic sulk in is/her future, but this is the "high water mark" sulk to date.

Epic usually refers to something very large, spectacular, or lengthy. These days the word epic is often used colloquially in a more low-key way. In this context the speaker is referring to the situation as an epic sulk, more or less meaning a great amount of complaining.