jmc

A sphere with radius $r$ has volume $\frac{4}{3}\pi r^3$, so a hemisphere with radius $r$ has volume $\frac{2}{3}\pi r^3$. The large hemisphere-shaped bowl has volume $\frac{2}{3}\pi (1^3)=\frac{2}{3}\pi$ cubic feet.

Let each of the smaller hemisphere-shaped molds have radius $r$. Their total volume, in terms of $r$, is $27\cdot \frac{2}{3}\pi r^3$ cubic feet, so we have $$27\cdot \frac{2}{3}\pi r^3=\frac{2}{3}\pi .$$Dividing both sides by $\frac{2}{3}\pi$ yields $27r^3=1$, so $r=\sqrt[3]{\frac{1}{27}}=\boxed{\frac{1}{3}}$ feet.