We used to do this years ago, only not in a barrel. We took a 4' tall piece of fence about 6' long, and connected the sides...like a barrel made out of fence! It's really been a long time ago, but I'm thinking we put some soil and compost and a little straw in the bottom and planted the
potatoes. When they grew and we could see leaves, we'd add straw on top of it. Don't remember if we mixed any soil or anything with the straw, but we just kept doing that over and over. We could just dig into it from the sides of the fence wire and pull out taters! It was really easy. Maybe someone else will post the particulars!