The primary use of a chairmaking kiln is to super-dry tenons. It’s nice to be able to set bends in a kiln too. This requires the kiln to be large enough to fit the bend while it’s in the bending form. Some bent parts have tenons, like a loop back bow, so ideally these tenons would be super-dried. My kiln is 2x2x3′ and that’s plenty big enough for two or three chairs worth of bends and other parts. Most folks can get by with something smaller.
A kiln needs to leak some air. Otherwise, you’ll just make a sauna in the kiln when you put wet wood in it. How much air flow you need depends on how much moisture is being removed from the parts. All my kilns have been leaky enough without needing to consciously vent them, but if your tight bends (loop, sack, continuous arm…) fail to fall out of the form in a day or two you may need to crack a door.
Holes in the top of the kiln work great for leg tenons, since you only want to dry one end of the leg. Stretchers, posts and arm stumps have to go inside the kiln since both ends of these parts have tenons. Wrapping aluminum foil or plastic wrap around the center of these parts will prevent the center from drying as quickly, both to prevent cracking and to keep mortices/prospective mortices from getting as dry as the tenons.
Insulation board and some tape makes a great temporary kiln. A major advantage is that you can tape it together when you need it and store it flat when you don’t.
Racks can be made of wood slats, dowels, metal shelving, old oven racks, etc. They probably should not be solid, so you don’t restrict air flow through the kiln.
My kiln has thin plywood sandwiching the insulation board to protect the insulation from regular use. It is built under my sharpening bench to maximize floor-space.
Curtis Buchanan put a heater in his basement closet and dries his bends in there. Old refrigerators/chest freezers with a heat source added can make great kilns (just make sure you drill enough holes to get some airflow).
My attic makes a great drying location for backs (I’m not sure it gets dry enough for tenons). Putting parts next to a woodstove or radiator in the winter can work well. An oven may work, if it’ll stay below 150 degrees. I hear microwaves will dry small pieces of wood extremely quickly. The entire state of AZ is considered a dry-kiln by some. Experiment!