You can often find pallet sheds for free on Craigslist so start searching then get to building!
Upcycling pallets to build projects is a sustainable activity, just make sure your pallets are chemical and nail free before using them.
Butch Bridges built this shed from pallets and there are a lot of people out there who want to save money and recycle and wish to find clean pallets for cheap that they can use in their own home improvement projects. Here is his project with more detailed information on how he built it.
A mini-industry has popped up for anyone with a pickup truck that is willing to find pallets and sell them.