In Excel 2007, I can right-click on a sheet tab and get the option to copy that sheet in the same or a different workbook. I get everything associated with that sheet, content, settings, variables, etc.

Is there anything like that in Visio 2003?

In Visio 2003, I can right-click the tab and select Insert. That copies the page dimensions, but none of the content including the layers. As far as I can tell, there is no way to copy anything to a different document.