Natural gas is brought into your home through the use of differential pressure. The main transmission line ties into your service provider which ties into your house. The transmission lines run at high pressure. When the pressure is cut through pressure regulators, it's then safe enough to be brought into your appliances.
To try and keep your eyes from glazing over, I will try and make the rest of this as simple as possible.
Field gas compressor stations (gathering systems that pull directly from wells) can run on either gas or electric. This is due to environmental permitting (combustion engines), power price/ availability, and the intended design of the plant (more or less complicated). The facilities that run on gas can survive for an indefinite amount of time in a natural disaster if the plant and pipe are not structurally compromised.
The larger refineries can only function without electricity temporarily running on backup generators. The emergency generators in the refineries I have been in only run heat trace, air compressors, the DCS /SIS (control system) UPS, and the lights. In an extended natural disaster like the one you just experienced, the emergency action plan would most likely be activated in order to bring the plant to a safe, stable state. For instance, the facility where I just transferred from uses way too much electricity to generate locally.
The nation as a whole has several underground storage locations that contain gas for just such an emergency as the one you are experiencing. The national grid is designed for withdrawal as needed to support extra demand or to float in case of a major outage of NG refineries. You can see the weekly storage report at this site: »ir.eia.gov/ngs/ngs.html
Without looking at a map of your local utilities, it's hard to judge how well your neighborhood will fare if a line breaks. Your utility should be able to answer that question for you. Pipeline maps should also be public record.
Hope that helps.