The distance from Washington D.C to Fayetteville, NC is 318 miles (512kms).
If you were to do the journey by car, this would take you around 5 hours 10 minutes, a little less if you are lucky with traffic.
Public transport will take you a little longer, between 6 hrs 30 and 8 hrs 30.
The quickest option by far is go to by plane. You can get a direct flight - there are 2 to 3 each day - and this would take you 1 hr 25 mins.
If you were feeling particularly adventurous you could cycle. It would take you 32 hours, so you might want to do it over a few days! You'd get to pass through some pretty great places along the way though: Prince William Forest Park has some beautiful scenery:
And William B. Umstead State Park near Raleigh would be a perfect last place to stop off.
(But you could also visit these places if you drove, of course!)