The only university in the United States that I know of offering a bachelors degree in Family History is Brigham Young University. While getting a degree in Family History would be an added plus, I can say that there are many professional genealogists with degrees in other fields and from all walks of life. What I do urge people to look at is the Board For Certification Of Genealogists. It is important to be aware and understand the standards that a professional genealogist should be working under. As far as other educational opportunities, there are institutes, webinars, continuing education classes offered with a certificate of completion, NGS Home Study course, and conferences.
Can you make a career out being a professional genealogist? There are a number of people I know of that do. It just depends what you want to do with your career. Some do research for clients, write articles or books, lecture around the country, teach, work for genealogy based companies (think Ancestry.com), become genealogy book sellers, or work specializing in some other niche like forensics.
Now for the questions everybody wants to know... money. Well, again it depends. This is the white elephant in the room that people will talk around, but never really address directly. With that in mind I can't give you a number figure. I just don't know. What I do know, is that there are people who are able to support their families and pay bills from working as a professional genealogist. I've never presumed to ask them how much. There are also some retired people I know that do not work as a genealogist full-time, who bring in "a little extra" and that suits them just fine.
Some of the factors to consider is where you live in the country. If you live near the D.C. area you have major archives there, which gives you an advantage. People need record pulls from those archives (NARA, DAR, etc.) all the time. Living near the Family History Library in Salt Lake City, Utah is another busy place for a genealogist. Salt Lake is also where Ancestry.com is based. If you have a niche or specialty that you become known for, it's another way to make money. Like I said... it depends.
So my fellow genealogy peeps, what do you have to add?