For relatively shallow rooted plants such as the cabbage family you would need 6 to 8 inches of soil. Like all plants many cabbage family members can send roots deeper than that but that is what they need. Other plants may need something deeper although most can grow well with that 6 to 8 inches of soil.

Are the bottoms open or closed? The link below is to a study about root development for various vegetables .. with charts and pictures. For example,'shallow rooted' cabbage roots can go down 2 1/2 ft. Ditto for beans. Tomato has more roots in 1st 3" than cabbage, but roots can go down 4ft. [my container 'toms' prosper in 3ft]

Here is a link that might be useful: Root Development of Vegetable Crops