Well, I definitely was able to copy my own 10.3.9 server every day for a few years without a problem, so I know it's possible.
So. I think it's likely that a file (logfile, database, something) is being written to aggressively when we try to copy it, and that fills the destination. Why that would prevent unmounting I don't know, but it's likely that's why the disk is filling. What file does it say was copied when the drive fills?