A couple of 5 or 6 inchers will work and from your description there will be 4' of the poles in the ground and the rest running up the sides of the cabinet.
24" holes 4' deep should be fine for a 10x10 on grade level. Here in Birmingham an engineered drawing is not required on anything under 20', the city engineer reviews the drawing and foundation work and will make changes accordingly. Some cities require one on everything.
Steel companies sell in 20-40' lengths but will cut them for you.
Good luck on your project.