A fully random 8 character password, using uppercase, lowercase, and digits, should still be OK today if hashed with bcrypt (or similar or stronger scheme).
I searched around and found someone posted some hashcat-ocl benchmarks done with a Titan X about a year ago, and it could do 14440 bcrypt hashes per second. So let's calculate how many "Titan X * days" are required to brute-force such a completely random password using that character set:
>>> (26 + 26 + 10) ** 8 / 14440 / 3600 / 24
175005
If you assume the best GPU today is twice as fast, and that someone has 8 of them in a rig, that's still 10 thousand days to search all possibilities, which means 5 thousand days on average until the hash collision is found.
For a typical user account, 5 thousand days of crunching by 8 of the most powerful GPUs on the market, is pretty good.
I personally go for 9 or 10 character passwords that include some symbols too, but are a bit less random ;)
Anyway, the biggest factor is to use a unique password for every account, because you can't guarantee that none of the websites you use didn't totally fuck up and accidentally put the original password somewhere, nor can you guarantee that they won't be hacked to the extent that the passwords are intercepted during sign-in.