You must complete "The Behemoth Showcase" in north part of the map before these become available or you can end up like me playing 15 h and not get your first barn find rumor.
After you complete "The Behemoth Showcase" (not straight away, after a few minutes in freeroam) you will be contacted via radio and informed about areal scans of barns and get and approximate location of the barn and you in game map will get a purple circle. If you do not get the call after completing The Behemoth Showcase, do Cross Country, Off-Road or Street races you have not done yet, or any other new activities.
Find the barn inside the circle by driving your car or flying drone and then approach it to trigger cutscene.
After that, barn find car will be automatically taken where mechanic will work on it and he will restore the car completely free if you wait (do races or freeroam) or you can pay him to get it instantly restored. The price varies depending of the value of the car and how much work on the car is left to do. You can check progress by visiting GARAGE, in BARN FINDS tab, by clicking on the car in question.
Achievement unlocks after the car is restored.
VERY IMPORTANT NOTE: DO NOT USE THE OPTION TO PAY TO INSTANTLY COMPLETE RESTORATION, IT MIGHT GLITCH THIS AND/OR ANTIQUE RESTORER ACHIEVEMENT!