As the software would need to be installed on each of the 4 machines, I would presume that you would infringe the software licence agreement. I am no expert, but to do it legally, youuld need 4 licences ! Anyone feel otherwise ? As far as the external drive is concerned, as long it had enough Gigabytes then I do not see why it should not be possible subject to the above caveat.
You could install Acronis on 1 PC and make a rescue CD. You can then boot the other computers from the rescue CD and create their images using that. Since the software is only installed on one PC then I don't think you will be infringing the licence. So long as you use different names for each of your images or put them into separate folders then you can store as many as will fit on your external drive.
The Jam I do exactly what you are asking about, namely store images from different machines on an external HDD. I would strongly suggest that you partition your external HDD so you have a separate partition for each machine.