Wow, so many!
For starters, Anathema is about as good as it gets for doom metal, along with My Dying Bride. I haven't heard a band evoke so much pain before...UNTIL I CAME ACROSS NOVEMBERS DOOM. Download the song "Awaken", WOW! It's all instrumental, but the guitar harmonies bleed sadness and pain like nothing I've heard yet. The closing line, "I will die someday; until that day, I embrace the knowing." hits me every time.
My list, in no certain order, is for both beautiful and depressing:
Novembers Doom -- Awaken
My Dying Bride -- For My Fallen Angel
Metallica -- Orion
Mazzy Star -- Fade into You
Lacuna Coil -- just about everything they play
Foo Fighters -- Floaty
Opeth -- A Fair Judgement, Harvest, The Drapery Falls, and When
Better Than Ezra -- Porcelain
Alice in Chains -- Nutshell, Rotten Apple, Would?, and Dirt
Anathema -- Fragile Dreams, Regret (the AHHHHH AHHHHHHAAAAHHHHHH harmony kills me every time), One Last Goodbye, Anyone, Anywhere, A Dying Wish, and Temporary Peace
Toadies -- Tyler
Acid Bath -- Bones of Baby Dolls
Toto -- Africa
Anthrax -- Black Lodge, and the hidden track on The Threat is Real...wow, one of the saddest songs EVER
Soundgarden -- Like Suicide, and I have to agree about 4th of July...great tune
The Police -- Every Breath You Take, and King of Pain
Phil Collins -- Take Me Home
Celestial Season -- Solar Lovers (the first album, it rocks)
Porcupine Tree -- everything they have ever released...they are truly amazing
Radiohead -- everything from The Bends
Crowded House -- Don't Dream It's Over
Doves -- Caught By the River
Bruce Dickinson -- Tears of the Dragon
Antimatter -- the entire new album
Agents of Oblivion -- Hangman's Daughter, and Endsmouth
I could go on and on, but I think these are good enough for starters.
For classical, check out Samuel Barber...ESPECIALLY "Adagio for Strings"...it's exactly what you're looking for when you think depressing classical.