Behold, the All Seeing Eye. One day, I gazed upon such a thing on the internets, and it intrigued me. I said to myself; Thou shall build thyself such a Thing! And in (approximately) 4 hours (give or take a week for parts), I have at last constructed such a Thing. And all the robots said "it was good".
In all seriousness though, my 360 camera consists of a few parts: (1) cheapo self focusing (IMPORTANT) webcam from Amazon.com, (2) plastic ends of a shipping tube (free), a small square of clear lexan (free), and (1) shiny silver Christmas ornament. Glue camera to one base, glue ornament to other base, roll lexan plastic into a tube, and glue both pieces onto each end of the tube. Viola! Insta-360.
Unfortunately the quality of the reflective surface for the ornament is very, very poor. I originally thought to use a concave circular car mirror I found at Wal-Mart for $1, but the FOV proved to be too narrow. Other options were unavailable due to expensive-ness.
Now to hook this up to a 'bot... hmm...