If you're really interested, staff and faculty have the Office of Mail Services, which comes around once a day to pick up any outgoing mail. The mail all has the department's cost center number written on it. Mail services them puts the appropriate postage (using a Pitney-Bowes machine or something similar, I imagine) on each piece of mail, charging its cost to the department's cost center.
The Georgetown post office on 31st street is about a 15 minute walk; this may come as a shock to people who didn't grow up in urban areas, but in cities, people actually walk places. In addition to the UPS in Poulton Hall, there is another UPS Store on N Street, just before Wisconsin, as well as a P.O. Boxes Etc. on Wisconsin between R & S Streets. There is also a Kinkos FedEx on M Street at Bank St., right next to SweetGreen.
The Wisconsin Avenue GUTS bus also takes you up Wisconsin, so you can go to P.O. Boxes Etc. from the first (Safeway) stop or to the USPS Calvert Station (across from Whole Foods) from the third stop.