look, if possible, try using an external USB disk and use the system backup option you have under Settings tab. It'll backup all PS3 games,trophies, accounts, etc. By what you're saying the HDD must be busted, or fragmented like mad (this happens when you turn off the PS3 unexpectedly, not using the power button and waiting to go to standby).
Once you're done, enter factory/service mode (a quick google search will tell you what is it), and try rebuilding the database.
FACTORY MODE SETTINGS:
• Restart System: Restarts the PLAYSTATION®3 System. (cold reboot)
• Restore Default Settings: Reset all system settings back to default. (erases user profile/system settings and builds a new one from ground up)
• Restore File System: Checks for corrupt/missing system files. (sort of scandisk, and will check/fix HDD if possible)
• Rebuild Database: Rebuilds the OS for the PLAYSTATION®3. (PS3 deals with a database system, caching most settings from there. Building a new database prevents excessive loading times)
• Restore PS3 System: Fresh restore; Deletes everything and starts from Scratch.
• System Update: Update the PLAYSTATION®3 System
now, careful because if you do important system changes while it's malfunctioning, the HDD (or the HPS partition) may be unrecoverable, thus, PS3 system asking you to re-format the drive to "fix" it, and you'll lose everything- So, it's important to denote, if you mess with these settings, do a full system backup first.