I used to raise ducklings in a cardboard box in the shower at our old place as they were safe from the cats there. OH was never too impressed though
Now they start off in an old 6'x4' shed which is lined with chicken wire inside to stop rats getting in and I can reach an extension lead there for a heat lamp. Once feathers arrive they usually move on in to a rabbit hutch for a bit until big enough and water-proof enough to move into a coop and run.
One thing I used to use are those big plastic storage boxes you can get from B&Q or similar. They are easy to clean out and with a bit of chicken mesh spread over the top, nothing can escape. Stick a desk lamp pointing in to it with a 100W light bulb and you've got a very cheap brooder box!