When I turn on my Sony CyberShot @120, the lens goes out and back in, and I get a message asking me to turn off the camera and turn it back on again. Then the same thing happens.

Does your lens go out and back in and so on, then you get a message "turn power off and on"? You probably dropped your camera or something was pushed or hit on the lens when it tried to open.
There are possible solutions here FIX #1-7, suggested by user typrice Try them first, if the do not work, try my risky FIX #8. It fixed my camera.
FIX #8. RISKY!!
- Turn the camera on,
- PULL THE INNER CYLINDER OF THE LENS OUT as much as you can when it is in the all the way out position.
- You will see a gap at the bottom of the inner cylinder of the lens. Keep pulling it out. It is just a cover that protects the mechanism of the lens itself. In my camera it was attached with a strong bond double tape (or glue). Eventually the cover comes off. Nothing broke.
- Take the cover off and you will see the actual mechanism of the lens. Be very careful not to expose it to any dust/dirt/small particles of anything. It's probably best to be doing this in medical vynil gloves and on a clean desk with good lighting.
- Take the "closure" off (the "doors" that close infront of the lens). Try turning your camera on and off to make the lens move. In my camera the lens got all the way out once and something poped!! and the lens stayed all the way out like it should.
- Put all the parts back in their places. Cover the lens... and hopefully enjoy your camera, like I'm enjoying mine.

I had this same problem with my pink cyber shot 120... I carried it to a local "camera doctor" and he said there was sand/dust in it. Maybe you should get it cleaned