At what distance(s), approximately, are high-voltage DC (HVDC) power lines more efficient than AC lines?
At what distance(s), approximately, are high-voltage DC (HVDC) power
lines \$\boxed{\text{more efficient}}\$ than AC lines?

 When the OP writes \$\boxed{\text{more efficient}}\$, I take it to mean efficient in terms of minimizing power loss
So, basically at any distance DC is more power efficient than AC.
DC is a more efficient transporter of power than AC due to skin-effect increasing AC power line losses and reactance causing AC regulation problems at the load. However, if you are looking for the reason why we normally transport power using AC then, there are a whole number of other things (unrelated to power efficiency) to consider.
AC is more efficiently \$\boxed{\text{generated}}\$ than DC and, AC transportation can make use of transformers to change voltages to suit the transport section.