Oh man... My initial pick is always Beethoven's Moonlight Sonata or Mozart's Requiem, the Lacrimosa, but those are well-known sad songs throughout history.
The saddest song I've ever heard is Dream Brother by Jeff Buckley.
It's mainly sad because of the meaning behind the song. Jeff Buckley was trying to convince a good friend of his not to abandon the woman who was pregnant with his child, since Jeff Buckley's own father had left him at a young age.
I also have to mention the song from the musical Into the Woods. I played Cinderella in the school play when I was a teen, but just recently, when I was going through a terrible situation, I heard this song again and actually sobbed:
Okay, one more, because I can't help myself...
Will Be Back by Im Sun Hae. It was originally from the OST of Scarlet Heart: Ryeo, and it basically played every time a character died. Even without knowing the lyrics, this song makes my throat clench.