Gravity is always pulling you down, but there are places in the solar system where gravity balances out. These are called Lagrange points and space agencies use them as stable places to put spacecraft. Nature is on to them and has already been using them for billions of years.

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I have the following question: Exist Lagrange points also for the four-body problem? i.e. exists point in a 4-body system such that one of them is (theoretically) stationary relative to the other three bodies ? (similar to the five Lagrange points from the classical three body problem).

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"Everything should be made as simple as possible but not simpler " (A. Einstein)

How large are these points? How many objects could you put at L5 and not have them run into each other?

At L1, the spacecraft are in orbits around the point. But how would that work at L4 or L5? Wikipedia (list of objects at Lagrangian points) states that the Spitzer Space Telescope is at L5, but the Spitzer_PocketGuide says it is moving further away from Earth in its trailing orbit.

I understood that the Lagrange points are "points" in a mathematical sense, i.e. for an object to stay indefinitely in a Lagrage point would need to have the size of a point (zero size)! So, for a spacecraft (or satellite) this point are unstable, hence the position of satellites is continuously adjusted by engines.

__________________
"Everything should be made as simple as possible but not simpler " (A. Einstein)

Typically a thousand kilometers from the Lagrange point the net gravity is perhaps a millionth of a g, so it takes very little energy to propel toward the zero point. If you do nothing, you will typically be in solar orbit in a few months, far from the Lagrange point.
Bodies such as Jupiter do shift the relative position of the Lagrange points (near Earth) minutely, so tiny amounts of station keeping energy are needed monthly or oftener. Typical craft shareing a lagrange point can be spaced hundreds of kilometers apart to reduce the possibility of collision at only a slight increace in station keeping energy. Some Lagrange points are more stable than others. I have also heard of orbiting a Lagrange point, but it is counter intuitive to me. Can someone explain?
My guess is there are no 5 body Lagrange point, perhaps no 4 body lagrange points. There are Earth-moon Lagrage points and there are Earth-Sun lagrange points. If there are Moon-Sun Lagrange points, they are likely very unstable. L1 (Earth-Moon) is closest about 40,000 kilometers from the moon approximately on the line between moon and Earth. Neil

So what is the advantage of putting a satellite in a lagrange point over just putting it in orbit? The way I understand orbit is that an object still doesn't have any engines on it that are continually firing to keep it in space, so it seems like the same idea.