Mineways is a free, open-source program for exporting Minecraft models for 3D printing or rendering. You don't need a 3D printer, you can easily upload and view (for free) and buy models using a 3D print service. See the gallery for some models people have made; some are for sale (normally at no markup), but I encourage you to instead make exports of your own creations.
You normally need the world to be saved on your computer, running Windows, Mac, or Linux. You interactively select a model from a world map, which creates a 3D print or rendering file. Preview this file with MeshLab or other 3D file viewer. Send it to your own 3D printer or a 3D printing service to make the model real. See the step-by-step guide or the two-minute introductory video for the whole process. Prices of 3D prints range from $5 to $500, depending on size - my mantra is "smaller is cheaper, cuter, and cooler". Or, just enjoy viewing and rendering your creation in a different way (but for rendering you should also consider jmc2obj).
Kids like Mineways, too. We've done two 2-hour workshops about 3D printing at Parts & Crafts in Somerville, Mass. - here are the presentation slides. If you're wondering, Mineways is a hobby of mine, I don't make money on it (just the opposite). In this interview and this podcast I talk about why I made it. Short version: for fun! My payoff is seeing what people do with Mineways, so please do send me pictures of 3D prints or renders or movies you make.
What should you do next to get going? Go to the quickstart page.