Thanks everyone. He has got it working right now. He followed some instructions he found on Apple's site, which included turning airplane mode off and then on again, and then restarting his phone. He stopped short of removing the SIM card because he was in a place where he really couldn't do that. The good news is that everything seems to be back to normal. Howver, if this happens again, he'll try removing the sim card.
Again this is a 5 which he's had it for 3-4 years. It's a work phone and he doesn't want a new one because the current "work plan" gives him unlimited data. A new phone would come with a "new plan" that does not include unlimited data. Personally, I think that's silly because a new phone would be so much better!