jmc

First, we can take out a factor of $z,$ to get $$z(z^4+z^3+2z^2+z+1)=0.$$We can write $z^4+z^3+2z^2+z+1=0$ as $$(z^4+z^3+z^2)+(z^2+z+1)=z^2(z^2+z+1)+(z^2+z+1)=(z^2+1)(z^2+z+1)=0.$$If $z=0,$ then $|z|=0.$

If $z^2+1=0,$ then $z^2=-1.$ Taking the absolute value of both sides, we get $|z^2|=1.$ Then $$|z{|}^2=1,$$so $|z|=1.$ (Also, the roots of $z^2+1=0$ are $z=\pm i,$ both of which have absolute value 1.)

If $z^2+z+1=0,$ then $(z-1)(z^2+z+1)=0,$ which expands as $z^3-1=0.$ Then $z^3=1.$ Taking the absolute value of both sides, we get $$|z^3|=1,$$so $|z{|}^3=1.$ Hence, $|z|=1.$

Therefore, the possible values of $|z|$ are $\boxed{0,1}.$