>>582528Welcome to the NHK has some feelsy moments, specially if you're a NEET, It's also a comedy, so don't expect the feels bit to hit you right in the front.
Serial Experiments Lain is a classic,
Madoka isn't really sad, but can get emotional. I've read the manga "Shigatsu wa Kimi no Uso", I heard it got an anime adaption, that's probably a decent bet. Speaking of manga: Nozoki Ana is a lewd one that has a few sad moments way later on. Probably better don't watch the anime of that one, looks like terribly cheap porn and is missing the charm
Oyasumi Punpun is probably the best manga you can ever read.
For anime films, you probably can't go wrong with Ghibli stuff, Spirited away is my favourite, but the saddest is probably Grave of the Fireflies.
5 centimers Per Second is pretty decent also
Cowboy Bebop is mostly space western, but there's some romance and tragedy in there, not really a tearjerker though.
Steins;Gate has it's moments,
Keep in mind most of those are not really non-stop sadfests, but rather generally decent series that also feature some sad moments (some more than others). If you are open to reading, I'd put Oyasumi Punpun on the top, if it absolutely has to be animated, Welcome to the NHK has my favourite feels.