Note: With recent changes to iTunes for Windows and Mac, this is no longer possible. Exposure records are still backed up as part of your device's backup though.
You can use iTunes to transfer saved exposure records from your iOS device to your PC or Mac. Details on that process are provided by Apple here: http://support.apple.com/kb/ht4094
Once the records have been transferred you can remove them from your iOS device. If needed, they can be put back onto your iOS device again by using iTunes.