The Hero's Journey is a framework that scholar Joseph Campbell came up with that many myths and stories follow. Many storytellers and story-readers find it a useful way to look at tale. (That's actually putting it lightly. Some people are straight-up obsessed.) Chris Vogler adapted Campbell's 17 stages of a hero's journey, which many screenwriters use while making movies. Vogler condensed Campbell's 17 stages down to 12, which is what we're using. Check out a general explanation of the 12 stages.
The story of Orpheus and Eurydice doesn't fit perfectly into the Hero's Journey structure, but we're giving it a shot. As the gross old saying goes, there's more than one way to skin a cat. Here's how we've diced up the story:
In the beginning, Orpheus and Eurydice are living Thrace, a mountainous region in northern Greece. They're in love and are going about their lives as any happy couple would: chillin' in meadows, listening to sappy music, and making doe eyes at each other. They even have a perfectly normal, uneventful wedding, where the most exciting thing that happens is that the god of marriage (Hymen) refuses to smile. Ho hum.
And then Eurydice gets bit by a snake and, um, dies. It's not exactly a "call to adventure," but it is the incident that disrupts the couple's happy lives. This event eventually leads to Orpheus' journey to the Underworld.
For a while, Orpheus doesn't do much except wander around Thrace, strumming on his lyre and crying about Eurydice. The thought that he should go down to Hades and rescue her hasn't occurred to him yet. Again, he isn't exactly "refusing" a call (because none has been explicitly issued), but he certainly isn't taking steps towards making his adventure happen.
In most adaptations of the myth, Orpheus decides on his own to travel to Hades. But a few versions say that the gods advised Orpheus to go down to the Underworld, because they were sick of all his moaning.
In order to get to the Underworld, Orpheus has to cross a few thresholds. First, he enters the cave that serves as the portal from earth to the Underworld. Then he crosses the River Styx (threshold between the living and the dead), and makes it past the gates of Hell, which are guarded by Cerberus, the three-headed dog. Yeesh.
As he searches for Hades and Persephone, the King and Queen of the Underworld, Orpheus passes by many of the sad, scary denizens of Hades. Lucky for him, he's able to charm them all with his beautiful music. His songs calm down Cerberus, who stands slack-jawed when he hears the music, instead of barking his head off, like usual. Then Orpheus enchants all of the shades, makes the Furies weep, causes Sisyphus to sit idle on his rock, and distracts the vultures from picking at Tityus' liver. He's basically a one-man band who tames all of these potential threats with music.
Finally, Orpheus finds Hades and Persephone. The authors of the myth don't say where he finds them, but the couple usually hangs out in Hades' palace, deep within the Underworld.
Using his enchanting music and persuasive speaking skills, Orpheus attempts to convince Hades and Persephone to let Eurydice go. According to Ovid, he gives a whole long speech, during which he reminds Hades that (a) even he, Hades, knows the power of love, having fallen for Persephone once, (b) Orpheus and Eurydice would both be dead eventually, so why not let Eurydice live a little longer?
Orpheus melts the hearts of Hades and Persephone, which is no small feat, considering they're the King and Queen of Death. They decide to let Eurydice go, under the condition that she follow Orpheus during the couple's trip back to the upper world (aka, the land of the living). Orpheus is also not allowed to look at his wife as they ascend. No matter what.
Orpheus and Eurydice travel towards the upper world. Orpheus can hear Eurydice's footsteps behind him but he isn't 100% sure that she's actually there.
Okay, usually in this stage, the hero confronts death for a final time during an epic battle: this only sort of happens in the Orpheus and Eurydice myth. As the couple nears the exit of the cave, Orpheus sees sunlight from the upper world. He is either so excited that he forgets his instructions from Hades, or he is suddenly overcome with extreme doubt about whether or not Eurydice is behind him. Either way, he spins around to look at her, which sends poor Eurydice plummeting back to Hades.
Orpheus reaches out to Eurydice, but he's just grabbing at air. So sad. He then tries to follow her back to the Underworld a second time, but the Ferryman on the River Styx won't let him pass this time.
Normally, this is where the hero emerges triumphant with his "prize"… but that doesn't happen in this myth. At all. Instead, Orpheus loses Eurydice forever. For a while, he wanders around Thrace, crying over his lost love. And then, one day, a pack of wild women called the Maenads show up and tear him limb from limb. They throw his head in the river and call it a day. So yeah, no prize.