Surprisingly, Brentwood doesn't have any antique shops of its own. You can drive south to Franklin or north to the 8th Avenue Antique district in Nashville. For a one-stop shop, consider an in-between option: Gas Lamp Antique Mall near 100 Oaks.
Mallory Station Storage is not your average storage facility. They also have the a furniture consignment shop, The Decorator's Den, under the same roof. Having one place to store and then sell furniture is such a great idea.