MacOS – How to Erase Hard Drive Mac System (this partition can’t be modified/unmount)

The Mac OS X base system image is not the cause of your problem. See What is the 'Mac OS X Base System' disk image on my 2011 MacBook AIr? for an explanation.

My best guess, based on the screenshot, as to why you can't repartition the Hitachi disk is that it's the current startup device. You need to restart your computer from an an external device—preferably a device that has the Lion Installer on it-if you want to reformat the drive. Otherwise, assuming you have a valid recovery partition, you can erase the Mac partition itself and reinstall Lion without an external device.

Perhaps Apple's Mac OS X 10.7 Help: Recover your entire system document will help you.

I can't test it but I think that if you erase the MBR (with dd instead of fdisk) and the partition table record you should be able to re-format you ssd. Boot from cd or external hd and open terminal

this code erase the MBR:

# dd if=/dev/zero of=/dev/disk0 bs=446 count=1

this one erase the partition table also:

# dd if=/dev/zero of=/dev/disk0 bs=512 count=1

If you erase just the mbr you could use testdisk to attempt to recover data (in case you don't have erased the disk label).