I have only grown one cabbage in my life and that was one my son brought home from school. It was a variety that under ideal conditions could grow to 40lbs or so.

I grew it in a container that holds 2 cubic feet of potting mix (about 15 gallons). I don't think it got more than 5lbs before it no longer liked the weather and the bugs decided it was tasty. I don't recall the roots being particularly crowded in that container.

I would think a 5 gal bucket would be OK for a variety that gets 5lbs, but I am certainly not a cabbage guru.

greengrass1, the spacing for growing cabbage commercially is 12" along the row and 24" between rows. I expect that 12" depth would be adequate for root growth. Cabbage is a leaf crop so adequate moisture and nitrogen would be the conditions to watch closely.