Not familiar with Quora in particular, but here's some bad things that can happend in general:
- You could suffer monetary loss, e.g. if credit card info was stolen.
- Your online identity could be connected to your offline identity. If you didn't want people to know what you have been writing or doing online, this can be a problem.
- Information you thought were private (e.g. private messages, emails) could suddenly become public.
- Your password could be exposed. If you reuse your password (still, lots of people do) other accounts could be breached as well. So a leak from a completely boring site without any interesting at all could suddenly cascade into other areas.
- Leaked information could be used to impersonate you, e.g. to answer security questions to take over other accounts.
- Leaked information could be used in spear phishing attacks against you, where you are fooled to believe the attacker is legitimate because she seems to know so much about you.
How severe the conseguenses are can wary from case to case. The infamous Ashley Madison leak lead to divorces and suicides. If you live in a country like China, having your political views exposed could land you in prison. But even the breach of a more innocent site like Quora could have unexpected and hard to predict consequences because of things like password reuse.
So if you have been reusing any password you ever used on Quora, I would change those right away.