WW: One tried and tested method for assembling steel without a welder is rolling seams and crimping them down.
It's kind of like rolling down the top of a bag of coffee. You put 2 pieces of steel together and fold the edge down over itself 180 degrees. Then you do it again, so you have a fully encloses seam joint. If you do it carefully (and there are many hand and bench tools to effectively make this kind of joint), you'll end up with something about as strong as the car's body panel, which is typically joined the same way (and then spot welded every few inches to keep the roll tight over 10 years of use).
Some of these joints, like the "overlap" joint in the top left, really need spot welds every so often if the metal will be flexed and moved around. Other joints that are stronger might not.
The bench tools typically are just a variation of a vise with a slot-machine handle on it instead of a screw. You put the metal into the thing and pull the handle to make an even 90-degree bend. Then you stick it in the machine a different way and it crimps it flat on itself. Then, if you want a double roll, you do it again. What you end up with is a coldwelded seam, thats pretty damned strong.
The hand tools are basically just big pliers with a wide head so that you can bend a few inches of metal at once. Only for small parts though, for long seams you really want the bench tool.
Welding sheet metal is a pain in the ass to be sure, but spot welding is a lot easier than trying to draw a bead down a piece of thin stainless steel.
I'm sure you know this already, but if you don't: to make a spot weld with cheap welding equipment, you drill part-way through the joint, and then just hit the hole with the welder and throw some material into the hole. It will melt around in there and fuse the metal surrounding the hole. It's much simpler on thin metal than a continuous bead. Since you arent keeping the current flowing all the time, you dont have to worry as much about burning through the sheet steel.
Lots of places make special drill bits to make the spot weld hole properly each time.
If you want to really do it easily, they make spot welder heads that are double sided. you just clamp down on the seam and give it juice, and POP you have an instant weld.
Either way, I think that a crimped seam is what you want to go for, and then reinforce some of the seams with strategic spot-welds.
I think this is definitely doable with some cheap tooling. You can pick it up from Harbor Freight, or wherever cheap shitty tooling is sold. You might find it in the paper, or if you know any guys who work in heating or cooling, their union should have a "for sale" bulletin board where people sell metal bending tools for ductwork.
Hope this is helpful.