The final episode of Cowboy Bebop, no matter how many times I watch it, gets me a little misty.

There are two moments in the anime Monster that do this to me as well. Once is the story arc after the library, which doesn't even have to kill anyone, yet after Grimmer delivers one line, my heart broke a little. Seriously one of the most depressing things in any form of media ever, trumped only by episode 72 of the same series. (Maybe 71, can't remember off the top of my head.)

If we are looking at games, the end of Lufia 2 is somewhat depressing. Two of your main character sacrifice themselves to save their home, leaving their child (Under a year old at the time) behind.

Xenosaga three had two deaths, both of which were depressing. One during the conclusion of the Yuriev story line, the other during the end.