Not only is it feasible to not have a car in DC, it's largely hopeless to expect to drive anywhere near the places you want to go. And, if you do, you will pay more to park your car than for a nice room at a good hotel.

DC has good public transportation (MetroRail and Circulator bus are great - Metro Bus not quite so much), much of it is very walkable and taxis and Uber are plentiful.

If for some reason you were motivated to drive to DC, I would advise you to park your car outside of town and use public transportation anyway.