I think the emptiness of it is what makes it depressing yet soothing. Speaking of soothing, Soothsayer also by Buckethead qualifies.
The melody loops yet I never grew tired of it. It plays in the area before the fight with Alice Twilight. I thought this was composed by Akira Yamaoka but I've also read the opposite, so I don't know.
This one gets me every time, because of the lyrics. Solaar's writing always gets to me. Unfortunately it is in French. But it's still a very good song.
I'll end this post with my favorite.
Yeah, I can't help but cry when listening to this.
Well, there ain't many songs that are sad itself, but instead, they make you remember something or they are related to something, and therefore make you sad (like in my case):
Brand New has this in spades, look them up. My favorite is Sowing Season, because the song just gets so good near the end, it just puts me to tears yet makes me want to rock out to it. Great when you're done being depressed and are ready to kick ass again.
The Queenstons - Our Special Place. There is no adequate and good version of it on youtube, so I'm just going to link the osu! map: http://osu.ppy.sh/s/19536
The song and its lyrics are at the very core of what sadness and depression convey. The feeling of everything being useless, and "this place [being] nowhere after all" and being "nothing [and] no one", compared to the rest of the world, and mostly, compared to the rest of the universe. Because in the end, everything is going to turn into nothing anyways, just by the law of entropy, and what our values mean and are now will not be of interest to anyone anymore when the heat death of the universe arrives, or similar scenarios. This song portrays that feeling of "Memento mori" in one of the most beautiful ways I've heard so far. Yet it's only half depressing, and also gives you reason to think why it might not matter that much for you that everything's going to end, and why "[hiding] away to overcome the fears that ruin all the days" is alright, why you can still live your life, even if you know that the end will eventually come for you. Bittersweet.