Internal (Apple SSD) hard drive problem
I do believe OS X Base System is the name of the mounted Recovery Partition image.

I would suggest completely erasing every last partition on your drive (except Recovery), recreating a Macintosh HD partition, and reinstalling OS X onto the newly partitioned Macintosh HD.

As for restoring your data, you can use Migration Assistant (/Applications/Utilities/Migration Assistant.app) to restore data from the Time Machine backup after you get a working system, or better, use the "Enter Time Machine" functionality from the TM menubar dropdown.