This achievement states 16000km/10000miles driven but it only unlocks once you've passed 10000 miles so don't keep flicking back to see if you've passed 16000 km on the world explorer.
A couple of things to note.
1) Do every other achievement first.
As daunting as this sounds, it's best to complete everything else as it all tallies towards your goal. This last achievement will take you an extremely long time if you decide to go straight for it without doing anything.
2) Do not quit a single race.
As annoying as it may be to finish a race when you know your going to lose, just restarting the race WILL NOT COUNT the miles you have just done. So if your pipped to the post by a computer or human player, complete the race anyway to count your distance.
I finished everything with 6000+km still to beat because I restarted all the time for the other achievements. Look at my achievements and see the time difference between the last two achievements won..... I'll say no more.
3) If you've done everything else.....
..... then your going to have to do single races on crossbay tunnel to cover the longest distance. Make sure you've got unlimited nitrous otherwise your just going to cause yourself more pain and set yourself a target eg. 10 races a day until you get the achievement.
And bizarely enough, going the wrong way round the track does not count towards your distance traveled so don't even think about it!
After a whole lot of pain, that's achievement unlocked!