The thing with backups is that they're just a copy of your data at a given point in time. Now, keep that in mind for a moment as I continue to offer another possible solution below (but read my caution afterwards).
It seems to me from your question you're just wanting a quick drag and copy solution. So, for instance, you could just make a copy of your Apple Mail data. To do this:
- Go to Finder
- Press and hold the option key while you click on the Go menu
- Select Library
- You can let go of the option key now
- Within the Library folder you'll see a folder called Mail
- Copy this folder (just right-click on the folder and select Copy "Mail", or press command+C)
- Navigate to where you want to save your backup (obviously another drive!)
- Now you can paste a copy (e.g. by pressing command+V)
There you have it, you've made a copy of all your Mail data.
A word of caution
However, you really need to consider if this will achieve the ultimate aim of what your boss wants? Remember I previously mentioned this was just a copy at a given point in time. This means if a month later you have a hard drive failure and need to restore your Mail backup, the data you'll restore is a month old (unless you've made a more recent copy - but that's something you have to remember to do manually).
This is where backup software (such as Time Machines) comes in. Time Machine automatically makes regular backups. Yes, the very first time TM makes a backup it will take a long time. However, after the initial backup, all subsequent backups take much less time. You may want to read: Use Time Machine to back up or restore your Mac.
Finally, there are other options to Time Machine, such as Carbon Copy Cloner and SuperDuper!