Yes! Any user with Admin permissions can disable data capture to stop all data capture. You can also selectively turn data capture on and off by domain.
Navigate to Settings > Data Capture and Privacy > Data Capture and change the Data Capture for [Account Name] toggle. It is important to note that no data will be captured until you turn data capture back on. Be sure to toggle it back on when you're ready to start capturing again.
Note: When you turn off data capture, sessions that were already underway may continue to capture for up to one minute, so you may notice a few more sessions populate even after you’ve turned off data capture.
What about Mobile Data Capture?
If you use FullStory for Mobile Apps and you want to disable data capture for mobile, turn off data capture by navigating to Settings > Data Capture and Privacy > Mobile Data Capture.