I store many things in cardboard boxes, and have no issues with bugs (but then, I am in an area with cold winters...other climates may produce different results).
You should be able to get free cardboard boxes from many places. I usually just go to my supermarket when they are stocking shelves and ask if I can take some empty boxes (inspect them for spilled food and for residual odors first!). I prefer boxes that held canned goods, or similar, which are both sturdy and odor-free. You can also call businesses that are local to you and ask if/when they set out boxes for recycling. My food bank always has a ton of boxes, because they get boxed donations; offices that use copy paper often have paper boxes; liquor stores often have wine or beer boxes free for the taking; and my "health/natural foods" type stores set boxes out for customers to use in lieu of bags...lots of ways to find free boxes!
Also where I live: there is free furniture on CL, as well as groups like Buy Nothing (on Facebook) or Freecycle (on yahoo groups). Several of my local thrift stores have very low prices and/or 1/2 price days, so that I can find furniture for under $5 (sometimes it's pretty crappy furniture, sometimes it just needs a little cosmetic repair). I realize this varies by location, and that you may not have a car, but things like plastic bins or wire shelves can be transported on a bike or bus pretty well. If you have a college near you, then dumpster-diving during the weekend when students move out (or for all of finals week) can produce shockingly good free stuff. Just to give you an idea of a few places to look, though YMMV. Good luck!