Yes, for the most part. The Fair Labor Standards Act (FLSA), the federal law that requires payment of the minimum wage, applies to all businesses that have $500,000 or more in annual sales. Even if your sales don't reach this threshold, your employees may still be covered if they work in "interstate commerce" -- commerce between states. Although this term might sound fairly restrictive, courts have interpreted it very broadly to include sending or receiving mail from out of state, making interstate phone calls, or handling goods that have moved interstate.
Even if your business is so small and local that it doesn't fall within these parameters, you may be subject to your state's minimum wage law. Some cities and counties also impose minimum wage requirements on businesses within their borders. For more information on minimum wage laws, see Nolo's article When Must Employers Pay the Minimum Wage?