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 3D print services. Check 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 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 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 probably be using jmc2obj).
The latest version for Windows is version 4.08, for Mac it's also 4.08, 32-bit only. Read the notes for the Mac - it's a bit tricky to use. For Linux, use WINE to emulate Windows. All versions work with Minecraft 1.2 and later. Mineways Version 1.15 works with Minecraft 1.1 and earlier, though I find it easier to just convert old worlds by loading them into Minecraft itself.
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 oppposite); in 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.