Originally Posted by
acinpdx
(I have to admit I'm having trouble visualizing your arrangement...)
beams aside, i think you might have trouble with the hammock idea because there isn't anything really bracing this structure diagonally...so you might get sway, or worse
For lateral restraint, you can:
1) mimic a pole building & have a deep set post (basically cantilevering up out of the ground & restrained by the strength of the posts); or
2) have a design that has either sheathing or diag corner brackets resisting the out-of-plane forces [you seem to be in this world based on your footing ques earlier]
I think what might determine your sizes for a small structure like this is more likely what looks good proportionally for timber framing (bigger likely better & showing differential btwn sizes of sub-elements).