As far as I know, you won't certainly NOT be hired, but
a law firm may opt not to hire you due to your
background.
It may depend on what state you live in, as some of
them require certification, which may make your
conviction more of a problem.
Theoretically, you could be a paralegal in most states
at a firm that would hire you, even with a felony
conviction. Most firms require an educational
background and some experience. If you cannot get
these due to your conviction, that might answer the
whole thing for you.
It's always worth it to call some law firms in your
area and just ask them if they would/are allowed to
hire someone with a record. They might be a little
hesitant, especially if it is illegal to discriminate
based on a criminal record in your state (the laws
vary), but you can at least see if its possible. What
have you got to lose?