Without more information it is somewhat difficult to provide an answer.
However I can say that from my experience an external hard drive is a lot slower than an internal drive. For maximum performance many Solid Edge users are switching to Solid State Drives for both the operating system and data storage.
So to answer what I can glean from your question - yes you can store documents and work on them from an external drive - but it will be slower.
Be careful with external hardrives. I have used them in the past with SE (USB2 USB3 and firewire) There have been instances when trying to save that SE will give an "unable to save because of network error" and the program locks you into a loop where you cannot save and have to close SE losing your changes. It looks like ST6 has a "save local copy" as a popup option to the network error message. The save local copy pop up has only worked for me about 50 percent of the time.
The issue is with the USB/Firewire drivers and or Win7 not being rock solid with the USB or firewire interfaces. For some reason.... occasionally Windows will drop the connection to the drive and SE doesn't like that.