Having different DJs is just for the sake of variety. I don't want to hear the same guy 24 hours a day, 7 days a week. Plus, there would have to be more than one to keep it broadcasting all day and night, unless they had a list of songs automatically playing late at night. And it can still stay a certain genre. Most radio stations do that.
It only needs to last a few years. After that they can replace it with the typical Fallout format, or replay recordings of the original broadcasts, which would be, at that point, quite substantial.
I think money is the only big issue. Ads would obviously ruin the immersion, so it would be quite costly. But I'm sure there would be plenty of college student willing to do it for minimum age. Although, hearing a bunch of teenagers all the time could ruin immersion as well.