I would want an 8x10 minimum with no bivy, but then I'm chicken that way :) Some of the hammock builders are designing tarps that have the ends cut to allow using them as doors. Zpacks makes a Cuben one that is 6.5oz :) http://zpacks.com/shelter/hammock_tarps.shtml
That is why the SMD Gatewood Cape exists. You get 360° protection, rain gear and pack cover for 11oz weight. You can turn it into a double-wall with floor and bug screen for another 7oz. You need 6 Ti stakes and one trekking pole, or SMD's pole, or a stick to make it work.
Bivies will give good protection under a small tarp, but they have their own issues-- cramped, condensation, durability, etc.