Before you gather your papers and head down to the nearest post office, you may want to take the time to call them, first. Not all post offices renew passports. If your post office does not, then hop on Google Maps and look for the next nearest and so on and call them and see if they renew passports.
You may be able to mail your passport to the National Passport Processing Center, given certain conditions.
National Passport Processing Center
Post Office Box 90155
Philadelphia, PA 19190-0155
You’ll need an old passport that is undamaged to submit with the application, it will need to have been issued within the last fifteen years, when you were at least sixteen, and was issued with your current name. You can get around this last part with certified copy of your marriage certificate or a government-issued document of your legal name change.