Many students considering a major or minor in Greek or Latin have this problem. The answer is "virtually anything."
Classics and Ancient Mediterranean Studies is actually a remarkably practical major or minor, as the training it provides will serve you well in almost any profession, and especially the humanistic ones. According to a recent study, classics majors achieve the highest means scores on the Verbal section of the GRE, and are near the top in the Analytical section as well; only one major scored even within ten points of Classics and Ancient Mediterranean Studies.
Another good way to answer this question is to see what other people have done after graduating with a degree in Classics and Ancient Mediterranean Studies.
Associate Technical Recruiter - International, Microsoft Corporation
Deputy Prosecuting Attorney, Kitsap County
Elementary Teacher, Puyallup School District
Teaching Fellowship, University at Buffalo
Medical Student, University of Pittsburgh
Another liberal arts college, Hamilton in upstate New York, indicates that their graduates are similarly engaged in diverse professions:
Policy Analyst, New York State Senate
Senior Vice President, First National Bank of Chicago
Senior Vice President for Business Affairs, ABC Inc.
Attorney, U.S. Environmental Protection Agency
Investment Banker, The Allington Company
Economist, IBM Corporation
Publishing & Media
Editor, Houghton Mifflin
Associate Director, Public Broadcasting Service
English Professor, Cornell University
Director, National Humanities Center
Classics Professor, Princeton University
Physician, U.S. Navy
Further examples come from a survey conducted by Princeton University of its Classics and Ancient Mediterranean Studies majors over a thirty year period; seventy-five percent (75%) of their respondents were engaged in law, medicine, business, or a variety of other occupations.
Still looking for some ideas? Drop by the department to read the pamphlet "Careers for Classicists" or check out some of these sites: