In a processor, they're using 0v and something like 0.7v or higher for a digital 1, not 0v and 5v. If they did that, you wouldn't have 8 bit microcontrollers running at 1.8v for example.
Yes, you could make 3 state processors... but you could simulate such a processor by simply using 2 bits at a time to store only 3 states.
You would need to reinvent a lot of math... as having only two states per bit makes it super easy to do multiplications and divisions for example ... ex /2 is just shifting a bit to the right... multiplying by 2 is shifting bits to left by one.
and btw you'd have 6,561 possible values. In binary you have 1 byte = 2^8 = 256 values, in ternary you have 3^8 = 6,561 and 3^16 is 43,046,721 bytes
... unless I'm so tired I'm not doing math well anymore.
having 3 states could be useful for some things, like the carry bit, overflow, dealing with negative and unsigned/signed issues etc etc but would really complicate a cpu design.