Can you completely live underground?

Defining the terms

What counts as "living" in Minecraft is very relative. Can you survive for an arbitrary time period? Sure, just put a beacon on the surface, set it to regeneration and wall yourself in closeby. No further action required. But is that fun? Would you consider that "playing a game"? I don't think so.

It's also a matter of definition what you consider "underground":

• Can you mine surface blocks as long as you yourself always have blocks above you? Then you could just walk under a manual flying machine.
• Can you go into the Nether? Probably, but what about the End? You could spawn underground, then only dig around and beat the dragon that way, then constantly switch between two tunnels to the surface to build a roof above the other one, then go into the portal. Technically still underground.

This can continue, "underground" is a pretty relative term and the only really clear definition would be "blocks above you". And I've played a Skyblock-like map recently while always having blocks above me, I wouldn't consider that "underground".

The (im)possibility of doing things underground

But let's still try to consider what is possible and what not. Let's define underground for this case as: "living in caves, generated or dug, with a considerable distance from the surface". Then it would be impossible to acquire:

• Things that only generate on the surface and can't be farmed underground. This includes dead bushes, totems and, because phantoms only spawn while you are on the surface, phantom membrane and slow falling potions, …
• Things that only generate in or around surface water and can't be farmed: hearts of the sea (and therefore conduits), sponges, prismarine, …
• Things from the End: End stone and products, dragon egg, shulker shells and boxes, elytra, dragon's breath, lingering potions, …

Of course some things could possibly be acquired, but it would be really difficult, like getting villagers (heal zombie), getting items from loot chests (only heart of the sea appears exclusively in treasure chests close to the surface and even they are usually buried) and so on.

Some more things depend on how strict or loose you define the rules. Also, many things depend on how long you are on the surface before you go down forever. If you create the world and instantly run into a cave, you're going to have a hard time at first. But if you first e.g. punch a few leaves and some grass, then it's much easier.

A lot of things are possible though, probably surprisingly:

• Growing trees, grass, etc. just requires light, not necessarily sky light.
• Chickens can spawn with zombie riders. But getting them to multiply is hard, because they behave differently than normal chickens.
• Fishing can give you many things, like saddles, name tags, leather, tripwire hooks, bamboo, lily pads, …