Pixar released a free version of their image-rendering application RenderMan. The "Free Non-Commercial Version" has no functional restrictions. The only limitation is that this version is strictly for personal use only. (Uploading finished videos to sites like YouTube is allowed, as long as you don’t monetize them.)
Currently a commercial 3D software like Maya, Katana or Cinema4D is required to use RenderMan. Pixar has announced (in the FAQ) that a version for the free and open source 3D software Blender is in development.
This means that (if you have the necessary skills and time) you could create your own fan Pixar movie at home using the same industry standard image-rendering application.