I don't have a yard, so I'd like to put sod on my balcony for easy access for my dogs. Is this possible, and what is the best/least expensive way to do this? Thank you

It is possible, but you should make sure that your balcony can support the weight. Ideally, you would need to put down a few inches of topsoil inside of a short wood planter for the sod to grow in. Remember that the grass will continue to grow, so you will need to keep it cut to an acceptable height, and also have it watered in the summer.