I am primarily trying to identify the types of compositions and approaches to composition that would produce such music, with any particular composers or pieces that might make good examples as helpful side notes. I should note that I mean classical music in a collective sense, not just the one era.
I'll come right out and admit that "ambient" and "atmospheric" are rather subjective terms. From a textural sense I'd say they refer to an unobtrusive rhythm (or such intense rhythm that the structure is no longer readily apparent), and a not necessarily harmonic structure but certainly not monophonic. I'd say these compositions would value "feel" over any sort of technical standard, without sacrificing the latter outright. The mood of the pieces should never feel especially urgent or powerful, yet still offer something of significance and some measure of emotional depth. If I can further clarify what I'm looking for please let me know how!
As a frame of reference, I personally consider some of the work of Stravinsky (certain parts of the Firebird suite), Satie (his Gymnopedies), and Reich (Music for 18 Musicians, his Counterpoints) to fit the bill. A good specific example is the 9 Beet Stretch which, in case you are unfamiliar, is Beethoven's 9th Symphony stretched to play over the course of 24 hours- it's essentially a neverending sound, something that could be constantly on in the background without distracting you. A better example (in that it's more concise and the various elements are more distinct, allowing the mood to be more obvious), albeit one outside of the genre, might be Radiohead's Treefingers.
EDIT: I have edited the question to emphasize my main point. To all who have solely provided examples, I apologize, but I do appreciate your input! I'm mainly looking for the types of compositions and the elements thereof; the examples were expected to be sidenotes.