What breed is he? You didn't say.
Just curious; is he registered or grade?
ANY horse can run barrels. ANY horse can do anything you train them to do. However, would I pick this horse to run barrels? No.
When I'm looking for a barrel horse, I want:
--Gradual shoulder slope
--Proportionate body hips/back/neck (although I don't mind if a horse is a teeny bit long in the back)
--Short cannon bones
--hocks low to the ground
--and a bonus if they have a pretty head
Your horse, as the others have mentioned, has his back legs too far underneath him, no hindquarter muscling or roundness, and cow-hocked. In addition, I also see that he has a very steep shoulder (typically makes for a rougher ride, so he may not be a good WP prospect), and long pasterns. His back is a tad bit long, but really not horrible.
Some of his issues can be fixed with regular exercise and conditioning, but some things will just need regular monitoring (such as his cow hocks).
Again, it doesn't mean that he can't run barrels. And he'd probably be fine on the local level if this potential buyer is looking for something just for fun, but I certainly would not consider him for a rodeo prospect or "big league" 4D barrel races.