Why were Aaron and his son only ones eligible for being a Priest? In ancient times, the only people who were offered a chance to become the priest were Aaron and his sons.

Would not this be unfair? There might have been people who were better suited for the position. It has been shown throughout the Bible that sons of people with the position of priest were not always good (Eli's sons, Samuel's sons, etc.).

What was the reason for this strictness regarding the position of being the priest?
What was the reason for this strictness regarding the position of
  being the priest?


I have come to believe that most of the commandments from God have a logical backing. And one simply has to image herself being present in the situation and position of God to understand it.

Suppose you were setting up a software company. How would you go about hiring good programmers? The first thing might be to put an advertisement for people with a computer science degree. Now why this strictness about having a college degree? Though it is possible to find good programmers who don't have one, these cases are rare and exceptional. A degree though just a piece of paper shows that the person has put substantial time, effort and money in systematic study of the field. And since - 


  Practice makes a [wo]man perfect.


Hence our strictness.

The Levites were a tribe that had a special job. They were not supposed to take up other professions like agriculture, metalworking or trading etc. Their life was made up of these things - 


Recite and memorize the scriptures.
Give sermons and practice oratory skills 
Meticulously copy manuscripts by hand and double-triple check them for copy errors.
Clean up and maintain the temple.
Coordinate social ceremonies and religious celebrations.


This is not to say that others could not excel in these things but that the probability was very less. Much less than in modern times because of these reasons - 


There were no schools, no books and no internet to gain information not relevant to your profession.
Life was much more busy. Cooking food on coal fire, cleaning up the barn and walking to the next village took entire day. This left little time for independent research for pursuing your ideal profession.


Now put yourself in the situation. You are setting up a temple. What would your advertisement read? Why your strictness?


  Matthew 22:14 For many are eligible but few are qualified. (Monika's
  Translation)