Odo doesn't eat and doesn't sleep - he just reverts to his liquid form and stays that way for a few hours. But that surely generates no energy. With that in mind, how does he get the energy that he needs to survive?
From the Memory Alpha page on Changelings:
Since Changelings do not eat, "Robert Wolfe used to theorize," said Ronald D. Moore, "that the Changelings pulled their energy directly from subspace (or some [other] quasi-scientific realm)." Wolfe's theory also stated that "Changelings use some sort of subspace 'pocket' to store additional mass during a morph."
This obviously non-canon recounting by one DS9 writer about another writer's harebrained ideas appears to be the closest to any canonical explanation of changeling metabolism, since the only on-screen references to eating are to explicitly say that they don't eat.
Directly from the webchat on September 12th, 1997 with unnamed questioners and Ronald Moore that Memory Alpha gets its quote from:
Questioner: Do you know what Odo uses as an energy source? I've asked this once before, but Odo doesn't eat food...and he sure doesn't use energizer batteries...
Moore: Robert Wolfe used to theorize that the Changelings pulled their energy directly from subspace (or some over quasi-scientific realm we have yet to establish) but it's not a question we're exactly foaming at the mouths to address on the show since it would lead us back into the Land of Technobabble.
And a chat on January 30th, 1998 also with Ronald Moore:
Questioner: And (sorry) how can Odo change his shape without altering his mass? If he turns into a rat, wouldn't he be an extremly heavy rat? What happens to his mass? Sorry about all the ?s
Moore: I have no idea. Robert Wolfe was a proponent of the theory that the Changelings use some sort of subspace "pocket" to store additional mass during a morph, but we've been careful not to address this issue directly since it's a good question without a really good answer.