Can you get prion diseases by eating anything other than brains?

Yes. Infectious types of prions have even been found in muscle tissue. But it's not that they could concentrate there adequately to cause infection. The entire biosphere has a substratum of prions with which all biological systems are in equilibrium. For infection to occur, this equilibrium must be tipped adversely which is hard to do. Feeding prion infected beef back to cattle can do, or cannibalism.
Possiby organ meats. Prions travel through the immune system and may be found in the spleen. Any organ that accumulates prions can be the source of an infection.

Related Questions

Do all prion diseases predominately affect the brain or also the spinal cord and nerves?

Prion. Prions are proteins that have changed conformation, so that the normal function of the protein inside the cells won't work. The prions' also causes other normal proteins to misfold and make more prions, spreading the process. The prions affect many cells in the body, but they particularly harm nerve cells in the brain, causing progressive rapid degeneration. They are not alive, and have no dna. Read more...