We can approach this problem by defining variables and building equations.

Let the capacity of Harold's old library be H, the capacity of Millicent's old library be M, and the capacity of their new library be N.

Harold has half as many books as Millicent: $$H=\frac{1}{2}M$$
The new library fits 1/3 of Harold's books and 1/2 of Millicent's books: $$N=\frac{1}{3}H+\frac{1}{2}M$$
Plugging in (1/2)M for H gives: $$N=\frac{1}{3}\left(\frac{1}{2}M\right)+\frac{1}{2}M$$ $$N=\frac{1}{6}M+\frac{3}{6}M$$ $$N=\frac{4}{6}M$$ $$N=\frac{2}{3}M$$
So the capacity of the new library is 2/3 the capacity of Millicent's old library, which is answer choice B.