Why do we need DME or RNAV when using VOR navigation above FL240?
The consensus of internet wisdom seems to be that no one knows for sure, but the most likely explanation is that before class A existed there was Positive Control Airspace (PCA) that started at FL240, and DME was a requirement in PCA. The main reasons for requiring DME were:


There's a limited number of high-altitude VORs so it isn't easy to use cross-radials from other VORs for fixes or position reporting; DME is a simple alternative
If there's no radar coverage then ATC can still manage lateral separation effectively using DME


That all seems plausible enough, but some sources out there say that PCA started at FL180, not FL240, e.g. this FAA history document:


  [VFR on top] was a principle [sic] factor in the [Grand Canyon] crash and led to
  "Positive Control Airspace" (PCA) FL180 and above. Because of the
  speeds of the aircraft involved, non-radar procedures were unwieldy
  (standards called for ten minutes between aircraft; some 80 miles for
  a typical jet), even with the development of aircraft installed
  Distance Measuring Equipment (DME) and the reduced separation
  available between DME equipped aircraft (20 miles).


FL240 is also where ATC starts using Mach numbers instead of knots (ATC orders section 5-7-1(f)) and the separation minima change (section 8-4-4). I don't know if that's for historical reasons or physical/operational reasons, but it looks to me like FL240 is considered the general starting point for high-altitude operations (14 CFR 61.31 requires a high-altitude endorsement for aircraft that can operate above 25,000' MSL).