Before you leave your vehicle at a storage facility's open lot, you should make sure that there's nothing of value in it. Electronics, jewelry or money should be removed and brought home. Also, the car should be covered so that no one can see inside.
If you want to store a clunker that's sentimental to you, a storage facility's open lot will suffice. However, though the space is safe from unauthorized access, such a slot will be exposed to the elements so this is something you will have to be OK with before you commit to storing your vehicle.
If you opt to put your car in an indoor storage facility, you should expect to pay a bit more. However given what exposure can do to your vehicle, the increase would be worth it as you will not have to deal with any damage the elements might cause.
If your car has been sitting for a long while at a storage facility, care should be taken before you start it up. Engine oil in particular, is a concern as it will have settled into the oil pan. As such, you will need to crank the engine a few times to get the oil circulating again.
If you don't want to leave your vehicle exposed to the elements but don't want to spend extra to store it indoors, then the storage facility's covered lot will suffice. Such a parking space will offer a higher level of protection against exposure and should help keep your car in good shape.
Before you leave a car for an extended period at a storage facility, it would be a good idea to perform a few maintenance tasks on it first. Its oil for instance, should be changed and it should be thoroughly cleaned before you drive it into your slot.