Many varieties of strawberries grow wonderfully in containers, including barrels, drums, or bags. Many seed companies and garden centers offer pouches or bags, complete with plants, for growing them vertically on walls and fences.
Personally, I like them in hanging baskets. With their green leaves, red fruit, and delicate white flowers, they're actually quite pretty. And when the juicy red fruits cascade down the sides of the basket, they are practically screaming for you to come over the pick them.
Like the growing bags, baskets save on patio space and keep the fruit up off the ground and away from mold, insects, and birds (mostly). Another idea is to grow them in planter boxes attached to railings or under sunny windows.
When planting in containers:
My husband brought home a 55 gallon plastic barrel from work. We cleaned it out, cut holes in a spiral pattern from head to foot. We then used a gardening cloth to line the barrel to help keep the plants intact. As we filled it with potting soil, we kept adding strawberries to each cut out. When it bloomed, it was absolutely beautiful. We had strawberries all summer. On top, we planted a large Daliah. I love my barrel and it takes up very little space.