We do not register firearms in South Carolina. Without a permit, you may carry your handgun, loaded or unloaded, in the glove box or center console of your vehicle, or in a container with an integral locking fastener, at your residence, at your fixed place of business (provided that business does not serve alcohol for on-premise consumption), or between your vehicle and one of those places. As active duty military, you may apply for a permit, with your orders as proof of residency.
I worked at NWS Charleston as a civilian contractor a couple years ago. I brought a 1911 with me every day. I declared my weapon at the gate every time I came aboard, and they would ask me to clear my weapon, fill out registration, and secure it in a locker while on board. I would pick it up on my home, using my copy of the registration as proof of ownership. If you live off base, you will probably be subjected to a similar procedure. I have no idea what the rules are for privately owned weapons storage in base housing.