Having more or less forgotten about the premise of the thread:
Rolling Stones - Love in Vain
Chicken Shack - I’d Rather go Blind
The Doors - Touch Me << I think I might possibly have been as stoned as this
Thirteenth Floor Elevators - Slip Inside This House or idly, and oddly Primal Scream - Slip Inside This House
I’ve never been a stoned a this ^^^^
Pretty sure I’ve put this in a thread or two already, but it bears repeating:
I’ve got no links:
+1 Adagio for Strings by Samuel Barber
+1 for Hallelujah , both Cohen and Buckley versions. Each brings something different to it.
“Amazing Grace” by Judy Collins, the version that sounds like it was recorded in a church with a choir for a background. (No, I am not religious).
“Mama I’m coming Home” by Ozzy. If you turn it up loud, real loud, so you can feel it fill the room.
“Sodade” by Cesaria Evora
“Without You” by Harry Nilsson
“Full of Grace” by Sarah McLachlan
Maybe not top tier achingly beautiful, but still pretty good:
“Heal Over” by KT Tunstall
“Don’t leave me on my own” by Chris Isaak (ok, pretty much anything by Isaak is sad but pretty)
And, don’t judge me, “Voice Within” by Christina Aguilera. Not sad, just a great voice IMHO.
Creep (radiohead) as performed by Amanda Palmer is funny as well as beautiful.
I’ll nominate a few more:
Annie Lennox’s cover of Bob Marley’s Waiting in Vain.
Verdi Cries by 10,000 Maniacs.
If You’re Ready (Come Go With Me) by the Staple Singers.
If I Were Your Woman, And You Were My Man by Gladys Knight & the Pips.
Who Knows Where the Time Goes? by Fairport Convention (featuring the late, great Sandy Denny.)
Somebody mentioned “Both Sides Now” by Judy Collins, but I prefer the more austere Joni Mitchell original version.
“I Love You Porgy” from Porgy & Bess, especially as done by Nina Simone
Billie Holliday - I Get Along Without You Very Well.
Addendum:
As for really Achingly Beautiful: Bach’s “erbarme dich” aria from the Mattheus Passion is pretty much at the top of my list.
Here’s a pretty good rendition: http://www.youtube.com/watch?v=ucg7l1g8G4g
I have to add one more - Summertime, again from Porgy & Bess, covered by Mahalia Jackson.
Man, youtube is dangerous. I’m gonna be up all night, I just know it.
Coming Up Close - 'Til Tuesday
Wise Up - Aimee Mann
You Do - Aimee Mann
All I Know - Art Garfunkel
Skywriter - Art Garfunkel
We’re All Alone (Unplugged version from “Fade Into Light”) - Boz Scaggs
Joey - Concrete Blonde
Heart of the Matter - Don Henley
Allegri. Miserere mei, deus
As it happens, I’m listening to one right now.
And, while it may not be some people’s cup of tea, I can’t appear in a music thread without plugging:
The best thing to come out of religion is the music.
-Edward Abbey
“Over the Rainbow” by Eva Cassidy
“Desperado” by Sheila Behman (from The Langley Schools Music Project)
Came in to post that one.
The Alan Parsons Project was consistently pretty damn good on doing hauntingly beautiful.
a) I Robot. Suggestion: crank it and ignore the cheesy video.
c) In the Lap of the Gods Crank this one to 11, boost the bass and middle-treble on the equalizer, and wish for better speakers.
d) Lucifer
f) And even the closest thing to disco that I actively enjoy listening to, In the Real World
While I love TSO, Christmas Canon can’t hold a candle to a Swedish Lucia. To watch this in person is magical.
That is great. 
Love that, downloading.
Here’s Radiohead doing The Smith’s Headmaster Ritual:
And Radiohead doing Joy Division and/or New Order’s Ceremony:
Achingly Beautiful:
I don’t suppose anyone else would like nine hours of Stereolab? No?
Well, I think these are very beautiful:
Television - Marquee Moon
Another one comes to mind, when coupled with the gorgeous lyrics:
Madredeus - Vem (Além de toda a solidão)
Translation from the original Portuguese.
A few more:
James Taylor Fire and Rain
John Denver:
Annie’s Song
Eagle and the Hawk
And of course Amazing Grace
Loreena McKennitt’s The Mummer’s Dance springs to mind.
There’s another song, “Patchwork” by Amberhawke, that I can’t find a link to anywhere. “Aching” doesn’t do it justice, though it applies, as does “haunting”. It’s a song about two young orphans who make their way by busking, and it spans from before their mother’s death to townsfolk finding their bodies in a snowbank. It’s beautiful–twelve string guitar and fiddle accompanying Linda King-Burdge’s husky voice that sounds like she’s crying the whole time–but the lyrics are brutal.