In terms of doing it on paper, it is impossible to trisect (or quintisect) angles with a conventional compass and straight edge.  And 180 = 2 * 2 * 3 * 3 * 5.  Of course you could be an anti-Babylonian rebel and create your own angle measurements on a power of 2.  

However, you can use a quadratitrix compass (essentially a moving plane device) for trisection (and quintisection) of angles, to satisfy the conventional degree measurement.  See  https://en.wikipedia.org/wiki/Quadratrix_of_Hippias for a very short discussion.

P.s.  Somewhat tangential, but of interest...see this video on origins of metrology for practical machining.  https://www.youtube.com/watch?v=gNRnrn5DE58