Under the NFA, there are no specifics on what type of storage device should be used to safeguard the NFA firearms. Most people tend to use gun safes that have key or combination lock. If necessary one can even use a separate lock inside of a safe to limit access to the firearms.
While it is permissible to use a locked drawer or closet to restrict access, one must realize that such a limited restriction on access will comply with the federal requirements but may create potential personal liability for damage done by the devices under a theory of negligence. If someone with a propensity to cause harm is able to easily circumvent the restricted access and you have knowledge of the risk, you can be held responsible under you state’s negligence laws for damages that result in your breach of duty by allowing them easy access to the items.
In other words while a gun safe is not a statutory requirement under federal law, it might be a good idea if someone who has access to your home might injure others through easily accessing the items.
Although Florida and most states do not have any specific requirements for storage, except if there are minor children in the home, you should verify this before using a limited security storage device like a closet or drawer.