How can I measure a droplets contact angle without a setup?
In 2010, Borguet et al. published in Journal of Chemical Education, DOI: 10.1021/ed100468u an intentionally simplified setup which may be useful as a reference.  A Sony Cybershot camera was placed in front of a sample scrutinized under diffuse illumination:



The additional lens shown is focussed on the drop (both for water and $n$-hexadecane, the authors recommend a volume of $\pu{5 uL}$; focal length of the length used equals $\pu{50 mm}$):.

With photos recorded like this



the contact angle determination was performed with the contact angle plugin of imageJ.  As Fiji, it is open source, free ware, and java based/OS portable.  To quote from their publication:


  For each easurement, the user must choose two points to manually
  define the baseline and three points along the drop profile (see
  the supporting information that features a set of instructions
  supplying all the details on the procedure used to make the
  measurement). The program then fits the profile of the drop and
  calculates the contact angle using the sphere approximation
  or the ellipse approximation. In our study, the ellipse approximation gave consistent results for contact angles $&gt; 40^\circ{}$. For
  drops with contact angles $&lt; 40^\circ{}$, the sphere approximation was
  used.


In comparison with reference data, the authors judge contact angles recorded as well as the subsequently calculated surface tensions as good enough for instruction / lab classes and research lab.