Well, I don't know about other states, but an unpardoned felony is not an absolute bar to becoming an attorney in VA. However, it will be a major, major issue in your character and fitness examination.
Here is an excerpt from the Virginia Board of Bar Examiners' website:
Conviction of a felony is not an absolute bar to taking the Virginia bar exam, but it is a factor which will be considered in determining whether a person can prove by clear and convincing evidence that he/she possesses the requisite good character and fitness to qualify for admission to the Virginia bar. The Character and Fitness Committee of the Board considers the nature of the crime, how long ago it was committed, the punishment, and positive contributions to society since the conviction. A pardon or a restoration of the person's civil rights is certainly a positive factor. For a listing of all the factors which are considered, see the Board's Rules.
It is possible that you will go through (and pay for!) 3 years of law school only to find that you cannot pass the character and fitness examination because of the past conviction.