It is entirely possible to have an operating system on an external hard drive, and whilst performance won't be spectacular as already mentioned here, it is far from sluggish if you utilise a high-speed interface such as USB 2. Of course you will need to have a BIOS that will boot from a USB device, but most do these days in all fairness.

Your best bet is to still have an internal drive with an operating system on it. That way, you can set up a dual boot scenario so it is not essential that you have the external drive present in order to use the machine.

It's not an entirely elegant solution but if you have a good reason to install an OS on an external hard drive, there is no reason why you cannot.