The Social Security Administration handles Medicare eligibility and enrollment. You can visit their website at www.socialsecurity.gov for specific information.
See also Social Security's FAQ: When should I sign up for Medicare benefits? which explains the enrollment process.
The local Social Security Administration office is located at 151 Corporate Parkway SE in south Aiken, off Whiskey Road behind the National Guard Armory & South Aiken High School. Their phone numbers are 800.772.1213 and 803.648.2356, and they are open Monday-Friday 9am-4pm.