why not just pull your hard drive out and send the tower back without the drive. i'm sorta amazed they want the hard drive back. usually these companies are pretty clear about you keeping it for data loss liability purposes.
though if it's an aser or gateway they will ask for the drive back too. it's part of thier wonderful world of what they call "customer service"
setting that asside to back up your data pulling the hard drive and sticking it in an old pc will work. if your toasted pc uses win7/win8 99% of your data will be located in c:\users\<username> folder
just copy the following folders
c:\users\<username>\desktop
c:\users\<username>\my documents
c:\users\<username>\my music
c:\users\<username>\my pictures
c:\users\<username>\my videos
if your web browser is IE you can get your bookmarks from
c:\users\<username>\favorites
if you have itunes on your pc for an iphone/ipad/ipod you can get all the data from
c:\users\<username>\itunes
if you have outlook on your pc, and want to save your email and the like, this will get a little more complicated because of the different versions of outlook, but basically for outlook 2010 and 2013your email data file is located-
c:\users\<username>\appdata\loca\microsoft\outlook\*.pst (copy the pst files located there)
your signatures are located in
c:\users\<username>\appdata\roaming\microsoft\signatures
your autofill email addresses are located at
c:\users\<username>\appdata\local\microsoft\outlook\roamcache\Stream_autocomplete_XXXXXXX
when you get your pc back, make your user account in windows, and just paste all those folders straight in. when you load outlook for the first time, create your email account(s) attach the datafile (its in the advanced email setup), make sure the sig and autocomplete files are in their origonal locations and go to town.