Use the fact that $(Y_1, Y_2, Y_3, Y_4)$ is equal in distribution to $\theta (Z_1, Z_2, Z_3, Z_4)$ where the $Z_i$'s represent the order statistics of four Uniform$(0, 1)$ random variables - this should be clear but prove it if it's not. Hence, for example, $\frac{Y_1}{Y_4}$ has the same distribution as $\frac{\theta Z_1}{\theta Z_4} = \frac{Z_1}{Z_4}$ which is free of $\theta$ and so ancillary. After applying Basu, we are victorious. The general theme here is that when you are looking at scale families you can easily get ancillary statistics by considering ratios.