Can I legally start a business at the age of 14 in Georgia?

I see no reason why it would not be legal, depending on the business.  For example, some businesses require licenses.  However, it is probably not practical without an adult's help, because minors have the right to void contracts.  For example, it is likely that no landlord will lease you office space because you would not be bound to the lease, while the landlord would be.  For the same reasons, you will have difficulty buying inventory, getting a web site designed, getting a Fedex account, hiring employees, engaging accountants, or acquiring any of the other services and/or materials needed to operate a business.  Moreover, apart from the age issue, you probably have no other business or credit history.  Banks will not be willing to finance your operations, landlords will want extra security, suppliers will require payment in advance, etc.