By all that is holy, someone actually made one. Or at least, a decent simulation of one.
All the successful computers ever made used binary, base two, to represent numbers. This is easy to work with, because computers are electrical devices. It either has current (on) or doesn't, and with many cells of this you could represent any number you care to think of, as well as the ideas of True (on) vs False (off).
A Ternary computer would operate on base 3. You'd have "Yes," "no," and "maybe." You could store larger numbers in the same number of cells. You could have one cell that branches based on its value. You'd have greatly increased complexity, because now you have to draw hard lines between what voltages are "Yes" vs which ones are "maybe." ("5V" vs "0V" are way easier to distinguish compared to "5V" vs "2V" vs "0V.")
Thanks to Awesome Geek Blog for pointing this out to me.