In the poem Song of Wandering Aengus, Yeats tells a variant of the myth of Angus Óg.
In Yeats’ version, Angus while fishing catches a small trout which magically transforms into a girl. The girl runs away and Angus vows to spend the rest of his life trying to find her. He then does so.
Yeats probably intends us to think about people who devote their lives to following a dream. In one sense, this is a good thing: it gives a life purpose. But of course, if the dream never materialises (it does not in this poem) some will consider the life a waste.
Yeats’ own life was devoted to two dreams: Irish independence and Maud Gonne. Both were partly successful, part failures.
In this poem, Yeats wants us to think about the quest, not the final destination.
It’s all about the quest, isn’t it? About seeking after a particular idea/ideal/idol. The man heads out because “a fire was in [his] head”, and he finds things to occupy himself, resulting in the catching of a silver fish. As he prepares to cook the fish, it transforms into a faery girl, who then vanishes into thin air. He is so entranced by the girl, who represents for him his heart’s dearest wish, that he spends years and years looking for that girl, hoping to meet her again, ending the poem as an old man. Not a defeated old man, but one who still holds hope that he will encounter the woman again.
If you wish to conclude that he is quixotic, chasing after a dream, then you may do so. If you wish to observe, perhaps, that holding onto dreams is tricky, because they are slippery things (like fish) that can wriggle away or otherwise vanish, then I think you’re probably onto something. There is something both hopeful and sad about the old man who has devoted his life to the pursuit of that girl, though. And there is something lovely and true about it as well, which is how this poem comes to be both hopeful and melancholy, an unusual pairing.