Allegri's Miserere performed by The Tallis Scholars (listening to it as I write this) certainly fits the bill. 'The Great Bear and Pleides' from Britten's Peter Grimes (and much more from that work too). 'Marche Funebre' from Beethoven's 3rd. John Dowland's 'In Darkness let me Dwelle'.
For me 'achingly beautiful' describes classical, choral and opera, but in a different way, Van Morrison's Astral Weeks, Tom Wait's 'Alice' and some Joni Mitchell stuff (try 'A Case of You' from her album 'Blue').