Possible reasons for not being accepted into a Familia:

1. You're not already an adventurer, as such you're "leveless" and "weak".
2. You're too low level.
3. You [i]look[/i] weak, scrawny, unfit, cowardly, like Bell Cranel, timid, overweight, etc.
4. You're too scandalous (some modest/prude Familia don't like the whole, 80% exposed skin-deal)
5. You look poor. Nobody wants beggars in their house, mooching on your resources.
6. The Familia are purists and don't accept [race] or people from [region].
7. You need to pay [sum] to become a member, like a membership fee.
8. The Familia doesn't go to the dungeon and has no need for combatants.
9. The Familia doesn't recruit, they only scout members. So unless you're chosen, you can't join.
10. Your [i]assets[/i] are too large and making the God/dess, Captain or other important members of the Familia distracted/jealous.

There's a few reasons, off the top of my head, that I can think of.