Like in cubic kilometers, what is the size of L1 "area of influence"? being unstable I guess that size can vary?
From this source I get:
The size of these islands varies. Each planet in the solar system has its own Lagrangian points. The islands of stability get bigger farther from the Sun and also for more massive planets. The ones associated with Earth are roughly 500,000 miles (800,000 kilometers) wide. The biggest zones (at least in the solar system) are Neptune’s; they are about 2 billion miles (3.2 billion km) across.
L1, L2, and L3 are unstable. If a Sattelite ventures too far from the Lagrangian area it will fall towards the sun. As you can see from the image below. This image came from the source linked above.
I'm not sure how much this will add, but the points are exactly that - just points. Gravitational bodies are often reduced to points for the purpose of calculations, such as, centers of mass and barycenters are also points. The Lagrange points are the same - mathematically calculated points based on the two objects in orbit around their mutual barycenter. The article @jmh posted says exactly that.
The L4 and L5 gravitational saddles where objects can have long term stable orbits and they can be very large, elongated regions generally in the same orbit ahead of or behind the orbiting planet, thought the size depends on a variety of factors. Jupiter's L4 and L5 saddle regions are enormous, for example. Larger across than the Earth's orbit around the Sun.
The L1-L3 points are different because there's no gravitational saddle where objects can remain in stable orbit. They still generate a region of low adjustment that I'm sure the size of those regions of low adjustment are pretty well understood by some at NASA. Satellites or telescopes that are put in L1 or L2 orbits require adjustment, but considerably less adjustment than a region in empty space away from a Lagrange point. They also remain in the same position relative to Earth, which has advantages.
The satellite region around L1/L2 that NASA uses could be defined as a kind of area of influence, but there's no exact measurement for where it ends as it gradually tapers off. Image below is not to scale.
The size also varies depending on which Lagrange point you're discussing. To my knowledge, we've only parked objects in the Earth-Sun L1 and L2 points. Moon-Earth Lagrange points have been discussed as parking spots for satellites, but never used to my knowledge.