Can an 8 gauge wire melt due to arcing on a 35 amp breaker
There is no chance of the entire wire burning up.   The wire failed at the terminations.  Unscrew the wire from the breaker and furnace, and closely inspect both sets of wire ends, and the screw terminals on which they terminate.  You will find either arc damage, or poor workmanship, or both.

Also check for any intermediate splices.   There is nothing wrong with those if they are done properly.   

It's also possble it's a failed breaker, but I wouldn't count on that.