I have a friend who investigated the possibility of this, he owned a small island off the coast of another larger inhabited island. Although he wasn't far enough off the coast to be declared international (ie international waters) he did tell me that in Canada he could issue commemerative coins (although not his own currency) and stamps.

If you have the backing and historical roots of a large country you can find a loophole:

http://en.wikipedia.org/wiki/Saint_Pierre_and_Miquelon

The trick is to make friends with your neighbors. http://en.wikipedia.org/wiki/North_Dumpling_Island

It's been tried with England (Sealand). And I did advice against it :)