Yes, you can definitely obtain a criminal justice degree if you have a felony on your record. However, whether you are able to obtain employment in that field after you graduate is a bit more uncertain. That will be strictly up to the employer. Many convicted felons have had difficulty obtaining the most menial of jobs with a felony on their records.
Yes, you will be able to obtain a degree in criminal justice with a felony on your record. When you are close to obtaining your degree, you will want to consult with a criminal attorney in your jurisdiction. You will want to consult with this attorney to see if you can have the felony expunged from your record. You will have a much easier time obtaining quality job offers if the felony is no longer on your record.