If meant for emergency use (e.g. hurricane), how do you cover it?

You don't. If you are in the middle of a hurricane/storm, just ride it out. Once the hurricane/storm is over, then start the generator up to power your refrigerator and what have you. If you plan to build a shelter for it. You will want to keep it 5 - 10 ft away from the house and have a shelter built around it. If you put it too close to the house, you may start smelling fumes from the generator.

You are on the right track; during hurricane Dolly three years ago I ran our generator from the carport under the kitchen window, running the appliance, and extension cords out the window. Worked fine (did this for six days)--Remember-- never refuel when unit is running. Protecting the unit from water is to safeguard the electrical connections. The generator can run while in a rain enviroment and not be harmed.

I built my own box, put vinyl siding, shingles and a large swinging door to remove for maintenance, etc. I also used louvers on the exhaust side with a vent on the air intake side. This also hides the ground wire and chain anchor for theft.