I got into higher education right after graduating with my bachelors degree. I started off as an administrative assistant, from there I went to work in the registrar's office as a transfer evaluator and that experience allowed me to become a counselor. While working in higher ed, one huge benefit is that you can go to that particular school for free. Thus, I worked on my masters degree in education while working as a transfer evaluator.
The key to getting your foot in the door, especially if you don't have a degree in a specific area like higher ed administration, counseling or education, is to possibly start within a lower position and work your way up. If you don't want to do that, I would reccommend just getting a masters in one of those subject areas and within your internship or experiential learning you will get the experience. Larger universities like George Washington love to higher people with that background.