It is possible, but you really have two main obstacles: obtaining your RN license and getting an employer to hire you.
To give you the best chance at accomplishing those goals, a criminal defense lawyer can really only help you with two things. First, you will want to see if your felony conviction can be reduced to a misdemeanor. If it was commercial burglary, it may potentially be reduced to a misdemeanor. Second, you will want to see if you are eligible for an expungement. If you went to prison, you are not eligible for an expungement.
Speak to a criminal defense lawyer by phone or in person for more thorough information.
Mr. Lounsbery has omitted the third -- and possibly most important -- obstacle, which is getting licensed by the California Board of Registered Nursing (BRN). Even if you completed nursing school and had a employer waiting to hire you, you might not be able to get licensed.
Since burglary is an offense involving dishonesty, your conviction can be considered evidence that you are unfit to become a nurse and you would have to show proof that you have been rehabilitated. The BRN will consider the circumstances of the crime, the length of time since your conviction, your performance on probation or parole, and what you have been doing since the conviction.
I agree that reducing the offense to a misdemeanor would be helpful, if you are eligible, but the BRN could still deny your application based on a misdemeanor burglary conviction.
However, a so-called "expungement" (actually a dismissal under Penal Code §1203.4) doesn't do much good. The conviction is still a part of the court's public records, which will also show the case was dismissed. Even if a judge granted your motion for a 1203.4 dismissal, you would still be required to disclose the conviction when applying to the BRN; if you didn't disclose it, that would be considered an additional act of dishonesty which the BRN could use to deny you a license.
Before you spend a bunch of money on lawyers and nursing school, why don't you set up a meeting with an admissions counselor at the school you would like to attend? The counselor could probably give you an assessment of your chances of becoming licensed as a nurse. (Beware of private "career colleges" that are operated for profit. Their admissions people often work on commission and have an interest in admitting everyone, even those who won't get a job. You'll probably get a more honest answer at a state-operated community college or University.)
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.