I often see classics students go on to pursue careers in law and medicine. Besides the fact that a significant portion of medical and legal terminology have Latin and Greek origins, classics as a discipline promotes critical thinking and mental discipline, which are certainly valuable traits to have as an aspiring doctor or lawyer.
Other fields that come to mind?
- Education: many classicists go on to teach at the collegiate or pre-collegiate level. Be aware that, if you want to be a college professor, it's going to require a PHD in classics / related discipline and the employment tends to be harder to come by and more unstable.
- Anything that requires solid writing skills, e.g. screenwriting, journalism, editing, etc.
- With additional education in something like Latin / Greek palaeography, you could pursue a career in manuscripts. Universities, museums, and auction houses all employ manuscript specialists.
That's just a short list.