Is it possible to build Jenkins jobs without using build numbers?
It's not possible to avoid setting the BUILD_NUMBER environment variable, but you can avoid using it. Try using the Jenkins plugin Build Name Setter Plugin. It adds an option in your build configuration to set the build name to whatever you want, and you can ignore the BUILD_NUMBER environment variable this way.

Consider 1.250.5 versus 1.250.${BUILD_NUMBER}. The former will be constant no matter how many times you build and the latter will increment by one every time you build. It's also possible to use other macro values in between ${} to make more interesting dynamic build numbers.