I did a search on this subject but the search only came up with "there's an example in the POV-Ray documentation" or "use the panoramic keyword with the camera". 

I tried to find this example but in vain. 

I tried the panoramic keyword but when using an angle of 360 degrees the picture doesn't look like I'm expecting it to look. 

So my question: Is there a way to create a picture with POV-Ray which is a 360 degree view of the scene taken from the point of the camera? 

If there isn't, one way I can imagine is to scan the scene with a rotating camera and put the pieces together afterwards. 
 

   camera {location <0, 0, 0> 
           look_at <0, 0, 1> 
           right x 
           up y 
           cylinder 1 
           angle 360} 

You can then rotate and translate the camera into the correct position. You could also try using cylinder 3 (instead of 1), depending on the sort of effect you want (the difference is explained in the section 7.4.1 of the docs). 

A note about rendering resolution; the above should be rendered with a ratio of 2*pi:1, eg. width 628 and height 100, or width 314 and height 50, etc.  If you want a larger vertical viewing distance, increase the size of the up vector and adjust the ratio accordingly, eg: to see twice as much vertically, use: 

   up y * 2 

and render at 628 by 200 (or 314 by 100, or 1256 by 400, etc).