In case of a harddisk crash where the system is running: Is it possible to copy
That depends alot on what type of HD crash you experience. If you just get bad sectors, copying most of the info to a new system might be possible, however if you head(s) take a dive into the platter itself the drive will become unreadable.
You may want to investigate some sort over RAID configuration esp. for a critical server. I just finished doing this to our mail server. It will give you pretty much instant (depending on type) recovery if something goes bad.
Keep in mind that RAID won't protect you from accidental user caused data loss, virus problems, the delete key etc. You should additionally have a rolling backup strategy for important data.
As for copying the directories to a new HD or machine, I accomplished this with a few minor headaches when I migrated a server to a new box,, it's not to involved but there are some little things to watch out for.