The new SHIELD TV supports external microSD flash drives and the new SHIELD TV Pro supports USB flash and hard drives.
If you have a Network Addressable Storage (NAS) or PC you can mount it to your SHIELD TV in Settings > Storage
To expand your internal storage for apps and games, your storage device will need to be set up as internal storage. Go to Settings -> Storage & reset -> Select the drive name, then select the "Set up as internal storage".
SHIELD TV (2015), SHIELD TV Pro (2015 & 2017) and the new SHIELD TV also include a microSD card slot for expandable storage.