Within the TV canon, The Q actually seem pretty limited as supposedly omnipotent beings go. On several occasions we see them die or suffer serious injury, make serious errors of judgement in predicting the intentions of 'lesser beings', they can suffer from evident depression, they're capable of removing each other's powers (when acting in unison) and generally they lack a sense of decorum and politeness.
On top of that, they're identified as having a limited IQ (albeit one far in advance of the average human) and a penchant for petty and vindictive acts of aggression. Q claims that the Q act out of a sense of "Superior Morality" which frankly no-one could possibly accept as true.
As regards their own perceptions of their weaknesses, we can see it in their own words;
QUINN: But you mustn't think of us as omnipotent, no matter what the Continuum would like you to believe. You and your ship seem
incredibly powerful to lifeforms without your technical expertise.
It's no different with us. We may appear omnipotent to you, but
believe me, we're not.
TUVOK: Intriguing. Just what vulnerabilities do the Q have?
QUINN: Always looking for the tactical advantage, Mister Tuvok. Very good. As a matter of fact, that's why I've come to see you. In a
way, our vulnerability is what this is all about. As the Q have
evolved, we've sacrificed many things along the way. Not just manners,
but mortality, and a sense of purpose, and a desire for change, and a
capacity to grow. Every loss is a new vulnerability, wouldn't you
Within the Trek EU canon, we see their limitations extremely clearly as Q (or other Qs) are variously trapped in the past, forced to re-attend Q university, turned into statues, turned into swords, turned into pure energy, forced by their peers to mediate planetary disputes, forced to marry Lwazana Troi (shudder) and generally treated as the comic relief in a whole host of comics, books and radio-plays.
In the IDW comic series "The Q Gambit, Part 6", we learn that the Q in fact share their "extra-dimensional space" with the wormhole prophets. After the Pah Wraiths take control of the wormhole, they begin to threaten the Q themselves.