On this page of the Forza Horizon 5 guide, you will learn what you need to do to unlock a Barn Find and pick it up.
In Forza Horizon 5, you can stumble upon barns while exploring the game world. When you approach the barn, you will trigger a cut-scene in which you see the Barn Find. The car will not be usable. You have to restore it first.
When you pick up a car, you need to enter the Workshop tab, which is available, for example, in your house. There you will find the Barn Finds tab. This is where you will be able to pick up your cars. If the car still has the clock icon, it means that its restoration is still ongoing and you can't collect it yet.
If you want to unlock a Barn Find quickly, you will have to pay with credits (CR). To find out the cost of speeding up the entire process, click on the car.