Is it possible to have emergency C-section without anesthesia?

Difficult. It's possible but realistically there is always local anesthesia available along with other meds which can be given quickly to help in an emergency. Truly emergent c-sections have been done with nothing (bad idea...), with IV meds (ketamine, versed, etc), or local injection (lidocaine, marcaine...Doesn't really work well either). If anesthesia isn't available (rare) it is possible but painful.
Yes,but. It is possible to do an emergency c-section under local anesthesia injected into the skin and muscles of the abdomen. This is not a pleasant experience for the mother and is done only in the direst circumstances.
Emergent C section. Sure. Whatever it takes to get the baby out alive and well.
Not and bcomfortable. It can be done with just local anesthetic....But you will be far from comfortable.