To reiterate: the "positive pressure" thing is dead wrong. Negative pressure is necessary to ensure odor elimination (filter and fan should be equally matched in CFM rating and a speed control should be used for such a small space). Positive pressure would cause odors to leak out any available opening...
I'd line that frame with Melamine-covered 1/8" Masonite/paneling. It should be about $10~$20 per 49"x97"x1/8" sheet. My reasoning is that it is durable, the plastic (Melamine) will already be white (no messing around with paint/films), and it'll give you one-piece surfaces in the sizes you need (easy to clean). Just caulk the frame before you lay it in (or use construction adhesive) and caulk the corners afterward (from the inside) and it should be sealed pretty near airtight. If your dimensions are correct this can be done with three sheets (but I'd recommend a seamless floor; that'll require another sheet).
I'd recommend installing the paneling on the inside, then insulating the box with styrene foam sheet ("pink board") to help stabilize the interior temperature. Exterior sheathing can be anything you want (to camouflage the box if necessary). I'd use 7/16" OSB sheathing with a Tyvek vapor barrier inside - sort of like building a "house within a room/closet" - because I have no need (legal or otherwise) to hide my cabinet. It may help (considering how your frame was constructed) to glue the melamine/paneling to the styrene, then glue/caulk them in place on the frame.
The 6:8 (exhaust to passive intake) ratio mentioned eariler is right on. I've posted a design for a light trap in another thread (it's rectangular so you'll need to do some basic math and finish carpentry - and know how to use a router - to make it suit your application). You can also alter it a bit to use standard-sized HVAC filters (the intake should have MERV 7 prefilter [outside]; MERV 11 final filter [inside]). These should capture most spores/pollen/pests and the scrap from the Melamine-covered panels would do nicely for the panels with some actual lumber (or plywood/MDF) for the frame. You'll want to at least prime (two coats) any bare wood.
This will only really "work" if the rest of the enclosure is well sealed, of course. Low-expansion "Great Stuff", low-VOC silicone caulk, and neoprene "weather stripping" are your friends as far as sealing things up go. Also; magnets/plates to help keep the door(s) closed/sealed are a good idea even if you use a padlock/hasp (which you totally should whether you live alone or not).
Speaking of which; I see no provisions for a door on your picture. How are you planning to get inside to tend the garden? I could see a "pull out" wall working in your frame but it will require careful measuring and a steady hand (a $25 electric "hot wire" foam cutter will be invaluable).
