The problem with defining 'fun' in any particular sense is that the concept is different for each individual person. One person can like shooters, another can hate them. Simply taking what the first person says at face value, and creating a simple shooter game, will only satisfy the first person.
However if you put, say, 5 people together and ask them whether they like being able to customise their character. If four people say yes, and one person says no, then being able to customise your character can be considered, in a fundamental sense, 'fun'.
This is where things start getting picky. In reality, you can define fun simply by saying 'anything the majority of people consider entertaining'. Yet that one person still finds it boring. This is where you look at two things. Who am I aiming for within my game? Once you achieve that, then you ask: what does the majority of the target audience consider entertainment?
So, for example, you decided to create a rogue-like. Your target audience is rogue-like fans, now you research what your target audience preferences in entertainment are. One of the biggest factors in a rogue-like for the fans is the variation within the randomness, the more they feel like every game is different, the more they enjoy the experience.
In reality, you'll never satisfy everyone. Just look at what you can do.