This just happened to my roommate while I was gone. If you paid for it, DISPUTE IT WITH YOUR CREDIT CARD COMPANY.
They lie and say that they're Apple support, but they aren't.
Anyway, my roommate's computer is probably full of malware now. She called them and talked to someone. She downloaded something that gave them full access to her computer. And then once they had control of her computer, they then told her they could fix the problem for a few hundred dollars. She said that she wasn't going to pay them when they said it was $350 to fix it, because her computer is 4 years old. But then he said she had a virus that would delete everything on her computer if she didn't, so she paid them.
He was still doing things when I got home and she told me about it. I'm assuming the computer is now a zombie computer. I do know they took a list of her contacts. I literally just got a Facebook message from them(never had one before). I turned her computer off. But unfortunately, I think the system is full of junk. It's probably going to need a total reinstall.
If you fall for this, you should immediately dispute the charge with your credit card company and cancel the card. After Googling it, it looks like other people are saying they get a whole string of fraud charges on it, a few days after paying them.