If they have restricted the max password length to 8 characters, it demonstrates that they are not correctly or safely hashing it into their database. So all of the data you have entered can be considered compromised.
It would be helpful to share the site so that others can avoid it.
If you use that password for anything else, change it everywhere else straight away.
If the site in question has any information about you, I recommend changing that information straight away to something random.
Afterwards, write to them and ask for all of your information to be removed from their systems immediately. Explain to them politely why you are doing this.
What danger can come from this?
All of the information you've entered must be considered compromised since they have clearly demonstrated a complete lack of understanding or care about the data.
Can this affect my email in anyway?
As Mirko says, if you've used the same password then you need to change it immediately. Otherwise, the main "danger" is from spam from compromised customer databases.
If my email is linked to social websites, will that be dangerous?
It isn't really dangerous but it may expose you to more spam. Spam can, of course, be dangerous in itself. You might also be exposed to cyber bullying or stalking depending on how vulnerable you might be to such things.