In a 4-player free for all game, would Elder Dragon hit every player's base and all 3 battlefields?

Yes. Elder Dragon's play effect targets "up to one enemy unit at each location," and in a 4-player FFA (FFA4/War, Rule 483), there are 7 locations on the board: 4 player Bases (each a Location per Rule 107.1.b) and 3 Battlefields (each a Location per Rule 107.2.b).

The ability creates a separate targeting restriction for each of those 7 locations. At your own Base, any units there are friendly, so you won't choose anything there. But at each of your 3 opponents' Bases and at all 3 Battlefields, you can choose up to one enemy unit and deal 1 damage to it — for a maximum of 6 targets.