-nervous to talk to people
-speaks slowly and quietly and stammers a lot around people he/she doesn't know as well as family members/close friends
-believes self to be inferior to almost anyone
-fears talking to people or family members or participating in social activities out of fear of coming off as an idiot
-is frequently preoccupied with the belief of his/her mental inferiority
-has difficulty keeping a job as a result of aforementioned symptoms.
-tries to find work that doesn't involve social interaction
-does best to avoid social interaction with people including family members/close friends (albeit not always)
-is happy being by him/herself
-believes he/she can be happy and content not associating with anyone for most of his/her life
-has felt this way for over a year

So, does this person have a serious problem? And what suggestions do you have for this person?

Yes, what you listed here does fit within the guidelines of avoidant personality disorder but they can fit other disorders as well. The best way to be diagnosed is through an experienced mental health counselor such as a therapist or psychiatrist, who will also be able to work with the person in helping him/her deal with this problem. There is a group of people with this problem who have started a website support group. Here is a link to their site.
Avoidant Personality Disorder Home ( AvPD ) (http://www.avoidantpersonality.com/)