Most sad, emotional songs tend to have at least some bittersweet moments, where the lyrics or music have a moment of hope, strength or dignity. Tears in Heaven by Clapton is a great, sad song, but there's a tender sweetness to it. Bonnie Raitt's I Can't Make You Love Me is great, too, but she sings with strength.
In that context, I'm not sure if I've heard a sadder song than Katy Song by the Red House Painters. It's less an artistic outpouring of emotion than the feeling of being intimately close to someone experiencing pure heartache. They nail that empty emotional space after a painful breakup when your heart, soul and mind feel painfully stripped of everything and you numbly grasp for some sort of meaning in the broken shards of memories you're left with. And from that void of despair, you feel the song move towards giving up on life and annihilation, like the final moments before death. It's a well made, lovely little song-- really! Katy Song