Why battery "sags" on my device and how to fix/workaround it?

Sometimes my device shows "No SIM card" status and "0%" battery and soon powers off (even if I manage to quicky connect it to USB). After powering back on it works more or less as expected. Why does this happen? How to fix the issue or how to force Android to continue if battery suddenly drops to 0% (as opposed of gradually become discharged)?