What wood?

How strong and how cheap. The terms are both fairly relative. Oak and maple are strong in all respects. Ash and hickory have good impact resistance. Douglas fir is pretty cheap.

Rockler's Wood Selections

First I'd ask what you intend to use the box for. Will this be inside/outside? A planter or a piece of furniture?

It will be outside. It needs to be as cheap as possible.

Use exterior grade plywood.

Yep. Reinforce each corner with 2 X 2 furring strips.

And galvanized nails or screws.

Hmm.. Exterior grade plywood would certianly do the job. If you are going to use it for something like a planter or a sandbox you'd have to go with at least 3/4". You could probably save some money by going with a plywood bottom and 2"x6" (8") Pressure Treated (look for non-toxic treating!) boards for the sides. If you buy the boards on 12' lengths they are usually a little cheaper and you can cut then evenly to the 6' length...

Compare the pricing at your local lumber yard and see which works out better for you.

Big, Cheap Box
Definitly Pressure Treated fir if the box is going to be in direct contact with the ground. The new type of PTF is of a non-toxic make-up, not like the old arsenic based ones. You should still wear gloves, when handling it. For direct contact you should use the Black PTF. The green PT board is made of Hemlock, I believe, and not as durable.

The height of the box isn't clear. If the box is 6' x 6' x 6', does it have to be made of wood. Corregated metal could be used for siding and roofing if you are building a large box. I would recommend pier blocks, PTF floor joists, and a plywood floor to start, and plywood or metal siding and roofing.

I highly recommend not using plywood, where ever the box touches the ground. If you are going to use plywood, raise it off of the ground.

You might also check local listings for "Used Lumber." Ideally Redwood would be one of the most desiralble materials for building an outside structure.

A wood Ive had excellent results for outside use (planters, etc) is cypress. Its readily available and is in 5/4 stock . (I hate nom 1"). i too would stay away from plywood touching ground, it will delaminate in a year or 2. Wolmanized woods have the Cu/As problem . Redwood or cedar are getting almost out of reach. Cypress is my first choice.
Stay away from "light" cypress, which is actually PAwlonia. This is good for painted furniture but not outside. Youll know cypress, its really heavy and it sort of stinks