Education in the Raj at that time and for decades afterwards had the basic educational qualification of “matriculation” taken at 16 or after 10 years of education and after that you had 4 years of college divided into two parts.
2 years leading to a degree called a “Faculty of Arts or Science (almost always referred by to by the abbreviation FA/FSc)
Then 2 years for a Bachelors of Arts or Science degree. You then could do a Masters (2 years), if accepted.
After FA/FSc but before BA/BSc a person could apply to and be accepted by a professional school like Medical/Dental, law, engineering etc or the military academies, they would then follow whatever that schools educational schedule was.
FA/FSc was a rather challenging course to pass. BA/BSc was considered difficult to fail.
The words BA (Fail) would not be something put in an advert in real life, except as a joke. It’s saying the person was too dumb to pass BA.