A quick search of apartments available in Georgetown from
Craig's List shows studio and 1 BR apartments in the $1000 - 2000 a month range.
You could probably live without a car, depending on how close you were to basic services (like grocery) and if your apartment and place of work were easily accessible by public transportation. They also have Zipcars/Flexcars (short term rentals) located near some metro and bus stations.