If someonelse could get your password db file (e.g. .kdbx file for keepass), maybe due to a hacking attack, how threaten is your security in fact? Of course this is a problem and you should change all your passwords, but how "dangerous" is this really? Is there any value without having the master password?
If you've picked a strong master password, they've effectively got an encrypted blob of data which they can try brute force your password from, but may never succeed (Assuming that the password DB format is free from errors).
However, unlike with a website password or other online service (Chat/ssh/etc), there are no limits on how many times they can try different passwords, or on what speed the password comparison runs at - they can send copies of the file to thousands of machines and try to brute force starting from different places to minimise the time to find the password. With a website, the web server is the limiting factor - even with thousands of source machines, it will only do comparisons at a rate it can support, or it'll fall over.
For a really good password, this doesn't matter - the number of potential passwords is far too high to try them all in a useful time (assuming for the purpose of argument that whatever data you have is probably not very useful to anyone in a few thousand/million years). Picking and remembering a really good password is harder though, so it's possible that it is actually a quote from a book or common phrase, which might be tried sooner than pure brute force would suggest.
If, on the other hand, you have a weak master password, they've got an easy to decrypt list of passwords and sites, and it probably won't take them long to get access.
Additional - based on @André's comment:
As André says, if the hack occurred on a device where the database is used, as opposed to one where it is just stored (main PC vs. backup server), there is a chance of a keylogger being used to find the master password. As a result, it would be recommended not to open any password databases on machines which are suspect until they have been completely rebuilt from a clean state. Most password managers try to prevent passwords being read from memory, although this is not 100% perfect, but again, this can be avoided by not opening the database on suspect devices (or ones which you don't have control over).