Would a Flurry of Blades and Elder Dragon in the enemy base kill my Baron?

Yes — your Baron dies.

Flurry of Blades deals 1 to all units at battlefields without choosing any targets, so Baron Nashor's "I can't be chosen by enemy spells and abilities" doesn't protect him — that protection only stops spells/abilities that choose or target him. Since your opponent controls Elder Dragon, that 1 damage is "their damage," and their passive makes any amount of it lethal to enemy units. Baron takes the damage and is killed in the following cleanup; Elder Dragon being in base (rather than at a battlefield) doesn't matter.

The only way Baron survives is if the damage is prevented (e.g., Unyielding Spirit) — otherwise 1 damage is enough to kill him. (RiftJudge FAQ #8684/#10171; Rule 323.5.)