good luck on the brew box project.
If the goal is how to get a decent barrel on a low budget, I believe Baybrewer has the right idea for Southern CA. Too bad no distilleries. How about tequila? That gets aged in white oak barrels a lot.
I can see the downside of a used barrel is they might be 50 bucks or less but are usually at least 50 gallons, which is a lot of cider to deal with at once. although you could get lucky and find a homebrewer selling a small one on craigslist.
If you go new, it looks like the cheapest you can get is a 5gal oak barrel shipped to you is about 200 bucks and it goes up from there, so I can see why you'd want to make something
For a small volume like 5 gal, you may not need the intrinsic structural integrity of a barrel to keep it together, but it will be tricky to keep it from warping over time with fermentation inside, humidity changes outside. Rectangular band are inherently weaker than round ones, but may be enough to do the job for 5-10 gal. It seems that evenly toasting a cubical box might be challenging. You might want to practice on a plywood one first.
+1 for pics of jimmyjet's barrels - sounds like you're close man. 3rd time's the charm