Living creatures have a metabolism, which is tied to game rules
Since there is no definition of 'living' in the rules, we fall back to the natural meaning in English...where there isn't a clear definition either.
Every fall I start my high school Biology class with the 'Definition of Life', and give the students a list of eight characteristics of living things. Over the next two lessons we discuss how these eight characteristics are guidelines to help us understand what we mean by life, but they are not hard and fast rules; not all living things have all eight of the characteristics, but things that have most of them are generally considered alive.
Some of these characteristics have nothing to do with 5e D&D. We don't know that within the game living things evolve, or have a genetic code, or are made up of cells. However, we do know that within the game living things have a metabolism, that is, they obtain and use materials to build their bodies and energy to run them. In the Monster Manual, living things are assumed to have a need to eat, drink, and breathe because of their metabolism which sustains their life - the biological processes of their physical body. Creatures that are not alive are specified to be of a different Nature.
For example, in the lore for Shambling Mounds we can read that (emphasis mine):
Despite its monstrous form, the shambling mound is a living plant that requires air and nourishment. Although it doesn't sleep the way an animal does, it can lie dormant for days on end before rising to hunt for food.
Whereas for Shadows:
Undead Nature. A shadow doesn't require air, food, drink, or sleep.
For Scarecrows:
Construct Nature. A scarecrow doesn't require air, food, drink, or sleep.
and for Water Wierds:
Elemental Nature. A water weird doesn't require air, food, drink, or sleep.
Thus, we can generally assume that living things are those which require food, drink and air, at least in some form and some amount to maintain the biological processes of their physical bodies, even if they do not follow the rules for food consumption given in the DMG. Anything which does not have these needs will have its atypical Nature called out, which tells us that it is not alive.
Mordenkainen Presents: Monsters of the Multiverse revises this somewhat, adding a more general collective trait called Unusual Nature, to spell out that all such creatures don’t need to eat, sleep or breathe.
You ask about "Constructs, Elementals, Fiends, Celestials, etc.?" As the examples above show, Constructs and Elementals have Natures specifying their lack of need to eat, drink or breathe, and thus they are not alive. Fiends and Celestials have no such unusual Nature specified. Presumably they need to eat, drink and breathe (although again, we do not know what or how much), meaning they have biological bodies they must maintain and are thus alive.
As a test of whether 'living' means maintaining a physical body with metabolic needs, we can look at creatures that don't have a physical body. While 'incorporeality' is not itself a feature, there are six creatures in the MM that are listed as having incorporeal movement. Of these, five are undead and don't need to eat, drink, or breathe because of their undead nature: banshees, ghosts, spectres, will-o-wisps, and wraiths. The last one is a demon - and demons, as stated above, are assumed to need to eat. However, this particular demon explicitly does not have a body. Of shadow demons, the MM says:
When a demon's body is destroyed but the fiend is prevented from reforming in the Abyss, its essence sometimes takes on a vague physical form...A
shadow demon uses its insubstantial claws to feast on its victim's fears, to taste its memories, and drink in its doubts.
Further, since its lack of a true body means that it does not have a true metabolism but more of a metaphorical one, the shadow demon has
Shadowy Nature. A shadow demon doesn't require air, food, drink, or sleep.