8'x8' platform 8' off the ground. If 4x4 steel columns, do I need bracing?
You could have a tremendous amount of dead weight on the platform without any trouble -- it's the liveliness of the load that causes problems. A single person weighing under 200 pounds can cause some sway in a 20,000+ pound bus by standing in the aisle and throwing his weight about, for instance. Consider the activities that will occur up there.

Have you considered a railing for fall safety? If the corner posts are chosen long enough to extend above platform level then they could support a railing; if the railing incorporates diagonal bracing or a shear plane (ie a sheet of plywood) it'll help stiffen the structure against swaying too (though not as much as bracing below the platform could do).

Does the storage of objects below the platform defeat the height of the platform? In other words, can the predators you're protecting against push the storage around and then use it as a step stool of sorts? If so then maybe you need to enclose the space beneath the platform. The enclosing walls could protect any bracing against being used by climbers.. but now the whole thing is starting to look a lot like a shed with a flat roof.