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The official requirements to get a passport at a passport agency are that you are "traveling internationally in less than 2 weeks or need to get a foreign visa within 4 weeks." You normally need an appointment, but at least some agencies are known to take walk-ins as an unadvertised policy, but you may be waiting a very long time. You'd need proof of your upcoming travel, such as a copy of your itinerary, but don't need proof of a life-or-death emergency.
There's no guarantee that it will be the same day though; they may have you come back the next day to pick it up sometimes, such as if you don't need to travel for a few days. But they will generally be able to get you a passport before your trip, barring any issues such as problems with your documents or verifying your identity.