It is impossible to do aikido without "ukemi", however it is possible and sometimes a revealing training tool to practice without "falling."
Ukemi does not mean falling. Its broader meaning is to receive energy with your body. Falling is only part of ukemi. The really sensitive "hidden" teachings of ukemi usually only become known, however, by lots and lots of relaxed falling at the hands of those that know.
Best scenario is...get this tons of falling experience when you're very young. As a "cradle language" if possible.
For those that begin this study later in life, it is a major part of the practice and a student must get to the place where the falling part of ukemi takes place with relaxation and proper form so that the mind/body system becomes sensitive to what is happening instead of worrying about getting hurt or what "ukemi" should be done with this specific technique.
Once you have gone past this sensitization stage, then practice without falling can be done with great benefit.