Barry's abduction scene, the general scenes of electronics going haywire, and to many, Roy's obvious descent into alien-induced madness. It's just so very deadpan and the setting so banal, yet his madness is profound.
Johnny Mathis singing "Chances Are" is just the cherry on the icing.
According to the making-of documentary, Barry's actor wasn't afraid of the abduction scene at the time. Not only was he 'abducted' into his real-life mother's arms, but Spielberg had told him all about what was going to happen a few hours before.
The scene where the government gases the escapees trying to climb Devil's Tower is very disturbing.