Most commercial DJs have very little, if any, discretion in what gets played. Most stations have a limited library that gets developed by program directors and through focus groups and audience testing, so they’re told to play just those songs. For example, one of my local commercial stations, WRVV-FM, has what they call their “Listener Programming Board”, which is just a voluntary focus group. When songs are perceived as being “played out”, they’re replaced with other songs. When songs are particularly liked, they’re played more often.
That said, some DJs will work wherever they can find it. Automation (which WhyNot asked about, and which I will discuss) is changing the nature of radio, so live DJ jobs are becoming harder to find. Some deliberately work for stations whose music they don’t like because it’s harder to burn out. Others work for stations whose music they like because it makes the job easier. From my own experience, though, I’m too busy doing other things to really listen to what I’m playing. It’s not just sitting down and spinning records, it’s filling out logs, finding commercials, and doing show prep and talking. Believe me, it’s harder than it looks. 
Okay. Automation. There are different systems for that, and different stations use it differently. At its heart, automation is any system that automatically plays music according to a set schedule. Some radio stations use automation overnight, because it saves on the cost of a live DJ. A DJ on an overnight shift may make $500 a week; the station can spend that same $500 once on an automation program. Some stations use what’s called “voice tracking”, which is basically when a live person records a show in advance. A DJ might work for 5 hours a week, but be on the air for 20. Again, it saves on the payroll costs associated with paying a live human being. There are, in fact, stations that have live humans only in the mornings and afternoons to give news, weather, traffic, and other information that can’t be recorded. It’s also possible to have a station with no humans in the on-air booth at all. (For an example of the worst-case scenario, here is some interesting reading material.) Finally, some stations just rip music into a computer and use that, just as you might rip your CDs into your computer.
In terms of automation, there are arguments both for and against. For management, it’s cheaper to use an automation system than it is to pay DJs. You also have a more consistent sound, less unpredictibility (which is no small concern in the age of huge FCC indecency fines), and greater control over content.
The cons are that it’s less spontaneous, more homogeneous, and offers a lot less local color. And as the link above shows, radio is supposed to serve “the public interest”, so not having anyone at the station can be dangerous in the event of a disaster. Economically speaking, automation also means fewer jobs for DJs as material can be recycled.
So, the short answer to WhyNot’s question is, yes, it is possible that the DJ can make himself a sandwich and come back when the light’s on. It’s just as likely, though, that the DJ is home making himself that sandwich.
Robin