Edit 2: My Nexus 5 doesn't boot up the OS or the Recovery, but it boots the Bootloader. The service people speculate that the power IC isn't able to supply enough power for the first two but it's enough for the bootloader. I have flashed a new 5.0 image using fastboot, and all fastboot operations report success. So I'm pretty convinced the internal memory is fine.
How do I recover my data? I don't know any fastboot method to do it.
Is there a way to connect the motherboard to an external power source so that adb works and I can read the data?
Is there a way to remove the memory chip and read it using a card reader or something?