It is possible to construct a structure in the backcountry. I hope your on private land. I don't know the laws in your area, but I imagine it wouldn't fly around here.
And since this isn't a lightweight idea, or the forum to be posting "how to build a shed" it's intriguing anyway. If I were to build a structure that would withstand the elements for a couple of months, I would use lightweight foam sheets cut so you could assemble panels. These would be lightweight, (well atleast lighter the wood etc). Depending on the size of your finished structure, and how to transport, you may need a truck. Personally, I would do a tent and check out Phat,s stove Idea.
But don't melt your shed down with a stove. Plastic and foam etc is very flammable!
Enjoy your next trip...