It depends upon the font that the symbol is defined in. Each font will be a slight bit different. There is a program available to create your own shp and shx files, but I cannot remember where it can be aquired from.
Now let me give you some good news. You don't have to redefine the degree symbol. With the advent of MText, you can specify a different text size for any character in a string as well as using different fonts in the same block of text. Simply highlight the text to be changed and then change the properties in the mtext editor window. You can easily scale individual letters or symbols, as well as include symbols and letters from other font files that are not included with the current textstyle.