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You have the right idea, but unless lumber and plywood is a whole lot cheaper where you are than it is here, or unless you can get some free wood and plywood, you can't make that budget. Quick calculation shows you need at least 10 sheets of plywood. I'm not counting what you might want for a roof. You'll need nails and screws. You'll need hinges. You'll need caulking. You'll need roofing material. You may want a window. If you want the place thoroughly sealed, you're going to want a window or at least a screened opening or you'll all suffocate at night. If you want the place to seal well against bugs, you might have to buy a real door instead of making one from plywood. You may want bunk beds or something, because 4 people lying on the floor of an 8x8 cabin are going to take up almost the whole floor. You'll want paint or stain, maybe, if only to brighten the inside.
You'll want some kind of foundation, because you can't set it right on the ground or it will be damp and the bugs will be at it in no time. I don't know what's appropriate in your area, maybe pressure treated wood will do, maybe it better be cinder blocks. If you have it off the ground and with a sufficient depth of eaves to keep the worst of the rain off the walls, it should stay dry
You can buy shed plans at places like Home Depot. You may even find them for free on the net. If you have any used building supply places around, you might check there after you make yourself a list of materials needed. If there's a Habitat for Humanity Restore, try there. With some persistence and patience you may be able to get at least some of your materials very cheap or free. If you have to buy new stuff, there really is nowhere cheaper than the big box stores unless you happen to have a friend with a sawmill.