jmc

The only way that the sum of absolute terms can be 0 is if each absolute value term is equal to 0. Thus, $$\begin{array}{rl}|x+y-1|& =0,\\ ||x|-x|& =0,\\ ||x-1|+x-1|& =0.\end{array}$$From the second equation, $|x|-x=0,$ or $|x|=x.$ Thus, $x$ must satisfy $x\ge 0.$

From the third equation, $|x-1|+x-1=0,$ or $|x-1|=1-x.$ Thus, $x$ must satisfy $1-x\ge 0,$ or $x\le 1.$

Finally, from the first equation, $x+y=1.$ Thus, the graph is the line segment joining $(0,1)$ and $(1,0)$. The length of this line segment is $\boxed{\sqrt{2}}.$