A company spokeswoman confirmed that a strike was under way but gave no details on its impact on production.
The union representative said the strike would lead to a shutdown of the refining units, adding that the process to shut them down began on Tuesday around 8 p.m. local time.
Workers are demanding wage hikes to cover inflation. Wage negotiations are scheduled for September but the CGT wants management to also commit to a bonus.
Exxon's Fos site has a refining capacity of 7 million tons per year, which corresponds to about 10% of national capacity, according to the company.
The walkouts were part of wider union efforts this week that have hit other energy companies such as state-owned electric power utility EDF.
At EDF, strikes have halted production of up to three gigawatts this week at a time when supply is already tight because several nuclear plants are under maintenance.
The industrial action in the energy sector comes as French President Emmanuel Macron is under pressure to alleviate pressure on household budgets from surging inflation.
Next week, the government is due to introduce new legislation aimed at boosting the purchasing power of families.
(Reporting by Benjamin Mallet; Writing by GV De Clercq; Editing by Tassilo Hummel and Deepa Babington)