Are Unarmed Strikes weapon attacks?

They are melee weapon attacks (basically a "physical" attack as opposed to a spell attack) but they are not attacks with melee weapons (precisely what it says, it needs a weapon).

This is a very confusing thing but if you can remember it then you should be able to discern what they work with (stunning strike and hunter's mark for example) and what they don't work with (sneak attacks divine smite for example)

EDIT : I listed hex as an exemple of things Unarmed Strikes don't synergize with even though they do and it's actually something I intend to use, I don't know why I made the mistake, sorry for the added confusion on an already confusing topic

What Zehal said. You have to look at the wording. "Melee weapon attack" (or Weapon Attack) and "Attack with a Melee Weapon" are treated differently by the rules. Melee Weapon Attack works with Unarmed Strike, Attack with a Melee Weapon does not.

From the Sage Advice Compendium concerning Stunning Strike (SAC is considered Official)

Can a monk use Stunning Strike with an unarmed strike, even though unarmed strikes aren’t weapons? Yes. Stunning Strike works with melee weapon attacks, and an unarmed strike is a special type of melee weapon attack. The game often makes exceptions to general rules, and this is an important exception: that unarmed strikes count as melee weapon attacks despite not being weapons.

As an example, if you multiclass with Fighter/Battle Master (or take the Martial Adept Feat) and pick up maneuvers like Lunging Attack or Maneuvering Attack it should be fine with unarmed strike. They use the wording Melee Weapon Attack or Weapon Attack. Unarmed Strikes are a special kind of Weapon Attack (they use Weapon Attack instead of Melee Weapon Attack with some Maneuvers because they can be used with melee or ranged weapons) But an ability that says Attack with a Melee Weapon it cannot be used with Unarmed Strikes.

Weird, I know, but there it is.