One might guess that /web/myf/log01 is a symbolic link instead of being a directory, but you haven't given us enough information to confirm that guess...
What output do you get from the command:
ls -ld /web/myf/log01
After the cd log01 , you could try the command:
cp /web/myf/*.db .
which should work to copy /web/myf/mykey.db into your current working directory no matter what directory you are sitting in.
It DOES work. If the source file exists and permissions allow for it.
The error message you see clearly states that the path/file does NOT exist.
Eliminating errors frequently is a tedious task which, to gain insight, best is approached step by step, of which one has been proposed by Don Cragun. If that either works or doesn't, you know the direction into which to dig deeper.