Hourglass networks handle features at multiple scales by using a series of convolutional layers that successively downsample and then upsample the input data. This process allows the network to capture and process features at different resolutions, enabling it to effectively handle tasks that involve complex spatial relationships. Additionally, the use of shortcut connections between mirroring layers helps the model to combine feature maps from earlier and later layers, further enhancing its ability to process features at multiple scales.