buy a hdd enclosure from amazon, 2.5" one btw
take the hard drive from the ps3 put it in the enclosure, transfer all the files to a pc
put hard drive back into ps3, send old ps3 away, new ps3 arrives.
find a tool that allows you to pack the ps3 files from the old ps3 into an official ps3 restore file
place newly packed files into an external hdd (NTFS format)
back up from external storage or something like that
all the game data is back where it should be