There are two flagpoles at the Kerbal Space Center. One is located in front of the Astronaut Complex while another is located at the Launch Pad. They show the default flag for the current save game.
The flagpole at the Astronaut Complex can be used to select the default flag for missions. These are not separate structures which can be selected like the Vehicle Assembly Building, etcetera; to access them you must travel to them after launching a craft on the Launch Pad or Runway.
Spoiler: Fun to discover on your own
Both flagpoles are climbable! However, you can't do a handstand at the top...
Kerbonaut atop the Launch Pad's flagpole with custom flag
The flagpole in front the the Astronaut Complex