Welcome to Android Central! Sorry to hear about the problem. What kind of data are you trying to back up? Is any of it on an external microSD card. If so, Unmount the card (in Settings>Storage), remove it, and insert it into your computer so you can copy all of the data there.
If the data is on the phone's Internal Storage, then is the phone still able to connect to your computer using the USB cable? If so, try to copy your files over that way. If you're unable to access data in Internal Storage due to the bad flash memory, then I'm afraid you're probably out of luck.
If all of your Contacts and Calendar events were linked to Google, then they should restore easily.