If this doesn't work then follow these steps: https://support.us.playstation.com/app/answers/det
I have not encountered Error 8002F271, but I did manage to fix the 8002F281 error.
I'm sure you have gone into recovery mode and pressed option 6 in the Recovery Menu which says "Update PS3." Then it said "Checking...." for a few minutes, then "Serious error has occurred, contact technical support. Error 8002F271" or something along those lines. What I did and what I recommend you do is these steps:
1. Remove the HDD from the PS3
2. Connect the PS3's HDD to a computer (I used an external harddrive's cable, I just unplugged it and directly plugged the cable into the PS3's HDD which connected to the computer via USB)
3. Format the PS3's HDD to either NTFS, FAT/FAT32 (They're pretty much the same), or Mac OSX Journaled
4. Download the newest update (The newest PS3UPDAT.PUP file from Sony's PlayStation website)
5. Put the PS3UPDAT.PUP file on a USB drive in the file structure PS3\UPDATE\(file goes in this folder)
6. Take your PS3 apart, to the point where you can access the battery attached to the motherboard, unplug the battery, then put it back and reassemble the PS3
7. Reinsert the PS3's HDD into the PS3 and plug in your USB
8. Go to the Recovery Menu, and press option 6 "Update PS3" like you did before
9. You PS3 should now ask you to format your HDD, when it finishes, it will install the update
10. Reapply the settings for your PS3 like you did when you first got it, then it will take you back to the XMB
11. Enjoy your now working PS3 and try to forget about the lost data &colon
Try Zeltryx's guide at your own risk. If you are in warranty this will invalidate it and you risk damaging the PS3 and yourself in doing so. (Take anti-static precautions and make sure there's no electricity in it or yourself).
In all honesty, as the errors indicate a corrupted hard disc, you'd be better off trying "Restore File System" from the Safe Mode menu, and should that fail to complete, try "Rebuild Database", then try it again.
I wouldn't recommend the above method as you're probably more likely to break it, than fix it trying to disassemble the console.