Jonathan, welcome to the Wolfram Community! There is already a post on 3D printing I recommend to read:

Creating a 3D mesh plot and exporting it to an STL file for 3D printing

C ormullion suggested a good possible solution. I still think we can at least try to print your original plot by increasing thickness of surface and joint points:



Now we need to export to STL and you can see the directory used for that (unless you specify a custom one) as:

To export and then check STL file is OK,  open STL file with your 3D Printing software or again with Mathematica: 

I don't think that will work out very well as it stands - there are disconnected pieces (although you might be able to print each one separately). I tried:

It's close to being printable, at least - but obviously not identical to the plot you want. If you print it in inches, it's 6 inches across. You will want to experiment with increasing the "detail" (PlotPoints -> n, for example). And the shape might require 'fixing' in an external program, because Mathematica doesn't always get the geometry right inside shapes (for example, there might be self-intersecting surfaces, etc.) In my experience it isn't always as easy as people say.