This is where I stay when in Key West at mile marker 10.
SeaSide Resort & Condo
55 Boca Chica Rd
Key West, FL 33040
They don't normally offer storage to the public but, I know they have space that is fenced. They might help you. Other than that, space is at a premium down there, especially secure.
If that doesn't work, PM me and I'll give you the name of a real estate person in Seaside that might be able to get you a lot at a reasonable price.
07 Dutch Star 4304
Celia 10, Alec 8, Bella (Greyhound), Isis (Abyssinian)