How old is the TC?
What model, A1xxx from the bottom if it?
Apple are telling people to store files on the TC so ring them up and say you were using it as you were told and now it is broken, what are they going to do?
They generally refuse to do more than swap to a new one.. the whole point of the TC is backup.. not file store.. so it has no way to backup and is considered a sealed unit.
What is dead btw? On a recent one, a dead hard disk is going to require some work to get files out of, if at all.
My advice would be to very carefully open the TC yourself and see if the TC works without the hard disk in it.. and plug the disk into a computer with usb to sata box.. you can get some recovery software if the disk does work.. otherwise it needs to go to a data centre. Then put it all back into the TC.. seal it up again.. and present it for replacement.. !!
http://www.ifixit.com/Device/Apple_Time_Capsule