No character traits

Sorry about the delay of this week's post. Before accusing me of being lazy, please allow me to say that I am that I've been trying to continue the robot-free streak I've been on. Moreover, the post I'm trying to write has undergone tens of changes, in my feeble attempt to convey the proper message since, no matter how many times I wrote it or re-phrased it, it ended up meaning a completely different thing altogether. In the end I gave up and posted the most recent version of it.

All characters have at least one trait, even if it's not mentioned specifically. As long as they do, think, say something, a trait will stem from there. Even in stories featuring otherwise mindless creatures, such as zombies, the character around which the action is centred must have a trait: be it that they're compassionate, merciless, active, humorous, they must have something that drives them through the story.

As a result, I was wondering whether it would be possible to write a story where the main character has no traits. Just like the woman in "B.D. în alertă" :„nici tânără, nici bătrână, nici înaltă, nici scundă, nici grasă, nici slabă” ("neither young, nor old, neither tall, nor short, neither fat nor thin"), they can't be defined by anything. I've spent months thinking whether there was already a type of character like this. After all, zombies are supposed to be mindless and robots are supposed to be soulless. However, in all stories I've come across, main character zombies had to have even an ounce of brain or soul and robots, although cold and soulless, were programmed to do actions that were either immoral and/or cruel or empathic and/or helpful. 
Would a main character that can't be defined through anything be dull? Would they be mysterious? Isn't being mysterious a trait in itself? How can it be avoided? Would it be possible to make a captivating story even with such a character?