Solution 1: Using the Warranty Properly - and a Warning

In the US the TimeCapsule has a one year warranty and if you purchased Apple Care you may have a longer hardware warranty.  Apple will gladly replace your broken device - BUT - they won’t copy your data to a new unit, nor will they swap out the hard drive to the new unit thus YOU LOOSE ALL OF YOUR BACKED-UP DATA.  Apple fails to help you because they treat the product as an “appliance” and thus will not open the device for repairs.  The “appliance” is sent back to a refurbish center where it will be given a new life. 
The Warning - Think of this:  all of your backed up data including passwords, company proprietary data, personal data, banking info, social security number, all that important stuff goes directly to Apple on the completely usable hard drive.  Technicians in the Apple stores do not destroy your personal data, thus hopefully whoever and wherever the refurb factory is they take care of destroying it (rather than using it).

Solution 2: You Actually Need the BACKED-UP DATA

This is the situation I was in.  I used the Time Capsule to actually backup my data and trusted it.  Here’s how you can recover your data (although not advised nor within Apple’s warranty or Apple Care program).  This is just a thought the may just perhaps work just fine!  Before going to the Apple store to request a warranty replacement remove the hard disk using the instructions in “Solution: Before it Breaks” then either replace it with a old hard disk (for the purposes of maintaining the weight) or just leave a void.  Since Apple stores treats this as an “appliance” they are not authorized to open these devices to repair them.  If you do not replace the hard disk with another one you may experience a conflict with an astute customer support representative that notices the weight difference - or you may not.  You can then keep this hard disk and replace the hard disk in your replacement unit to regain access to your data (again following some of the instructions in “Solution: Before it Breaks”). 

The Warning - Apple does not warranty the Time Capsule if it has been opened, so you must hide this fact by resealing the device and replacing its rubber bottom.