Word for a desire that you can not do anything about [duplicate]

"Pipe dream" encompasses the first requirement (that the word or phrase must define a desire or wish that is improbable to achieve), but I don't know that it fully satisfied your second component (that the desire or wish in question be intrinsically unattainable)

If you need a single word only, "chimera" means basically the same thing, but is probably less widely known.

I would use "yearning". In all examples I know, it has been used in the sense that it is a longing that carries the connotations of "intense", "unsatisfied", "never to be satisfied" or "very unlikely to be satisfied".