WASHINGTON—After fall workouts in September 2005, players from the University of Virginia baseball team often trekked 120 miles north on U.S. Route 29 from Charlottesville to Washington. They embarked on that journey to watch Ryan Zimmerman, the first draft pick in Nationals history who just three months earlier had battled alongside them in the NCAA tournament.

As Zimmerman’s friends traveled up the road, they noticed something outside the window that left them stunned: freshly minted billboards with Zimmerman’s face plastered...