How do you count 5 calendar years for social security tax exemption? I was on J-1 from 08/14 - 05/15 and then on F-1 from 08/15 - present (a few times out of the country)

You are considered as a resident alien from January 1, 2019 if you satisfy the Substantial Presence Test at that date.

Yes, but before you even do the substantial presence test, how do you count the 5 years of exemption? If I entered in August of 2014, do I count the whole 2014 as one calendar year? 

You can be a resident alien after 5 years or part-years in the US. Those 5 years are 2014, 2015, 2016, 2017 and 2018.