Another option is to store it inside a non-descript waterproof container. [google][amazon] Maybe hide it in the bushes, and have it connect to computers inside via Wi-Fi?
If you want to use Starlink for emergency purposes only, you'll want to familiarize yourself with their policies on Activation, Roam service plans, and Pausing service.
Activation for new hardware is often required, even if you pause service immediately afterward. [1][2]
The Starlink Mini itself is very weatherproof, so it can be mounted on the outside.
Most manufacturers of plastic hard-cases state that the case must be cracked open before powering the unit up, since it can build up a lot of heat inside.
Pros: Antennas can stay inside the enclosure, which simplifies weatherproofing.
Cons: Heat dissipation is a real problem.
Metal enclosure
Pros: Heat dissipation isn't a problem.
Cons: Antennas have to be placed outside the enclosure, which complicates weatherproofing.
The Starlink Mini is already weatherproof (it can be mounted on a pole or on top of an RV, as-is), so you don't have to worry about that as much. But you do have to deal with cable entry points.
Open your home's window and stick a pole-mounted Starlink Mini out
Presumably you'd only do this for a limited time, during emergencies.
Pros: This makes it much easier to provide power and to connect it to your home's network.
Cons: Landlords / condo building associations may not like this. You may want to pull it inside when you leave, so you can close the window, unless you can guarantee that rain won't come in through the window.