I am playing a life gain deck and I find that if an opponent plays Erebos, God of the Dead my deck is ruined because of its effect "your opponents can't gain life." If I run Aegis of the Gods, whose effect is "You have hexproof", does that mean Erebos' ability is negated, or because it doesn't actually say "Target opponent can't gain life" Aegis' effect doesn't actually work?
Aegis of the Gods will have no effect. You still won't be able to gain life, for the exact reason you suspect. From the definition of Hexproof:
702.11c "Hexproof" on a player means "You can't be the target of spells or abilities your opponents control."
Since Erebos' ability doesn't say 'target' anywhere, it doesn't target you, and Hexproof has no effect.
Alternately, you can find a way to remove Erebos from the battlefield. Exiling is one way. Erebos can't be destroyed, since Gods have Indestructible. Have you considered cars such as Deicide or Banishing Light?