This past Sept I spent two weeks hunting out of a tent in weather that was producing rain and or sleet about 60% of the time. I used an old gun case that was made by Browning back ??when??. It is made of what you would call oil cloth the same as an inexpensive oil cloth kitchen table cover. IOW, not actually oil but a plastic coated facbric backing. Very light and the rifle slid in easily. With that and an automotive vacuum line plug over the nipple, things stayed pretty dry. Fired, cleaned and reloaded the gun end of first week and it went off properly. Last day of hunt, same thing, went off without a hitch.
Don't know where you would find one of those cases, but you could make one pretty easily with a table cloth.
Other ideas here are excellent and superior in many ways, but the big advantage of the one I used is it is very light and absolutely water proof.