Suppose that we mark five arbitrary points, anywhere on Earth.
Is it then always possible to find a point in outer space from which you would see at least 4 of these 5 points?
If we assume that Earth is a perfect mathematical sphere, then
the answer is no.
Put one point into the Northern pole, one point into the Southern pole, and three points onto the equator that form an equilateral triangle.
(1) There is no way to simultaneously see Northern and Southern pole (you would have to move to infinity for this).
(2) There is no way to simultaneously see all three points on the equator. Otherwise you would see the full equator (and you would have to move to infinity for this).
(3) Hence you can see at most one of the poles, and at most two of the three equator points.
Not always possible to observe $4$ points. BONUS: Not even for $6$ points!
For example, place $4$ points on the equator so that they form $2$ pairs of antipodal points.
Now, we can only observe a maximum of $2$ of these points at a time from outer space.
As an aside: we can only observe a maximum of $1$ point from outer space if our line-of-sight is perpendicular to one of the antipodal-pair's connecting diameter.
Add a fifth point point and we can only observe a maximum of $3$ points.
BONUS: Place the fifth point on the south pole and a sixth point on the north pole we still can only observe a maximum of $3$ points.
Since this puzzle was tagged [lateral-thinking], I am compelled to submit this answer:
Yes, because mirrors. No matter where the points are placed it will always be possible to use (appropriately positioned) mirrors to see all 5 points (or more) simultaneously.
Yes because we have two eyes. All we need to do is extend the organic connection between them and our brain, enough so that we can see two hemispheres of earth simultaneously. (This would require a whole diameter of earth between our eyes, and some impressive parallax)
One more lateral-thinking answer:
Yes, all 5 points should be visible eventually, because the question does not state that they need to be viewed at the same time.
Given that the earth rotates about its axis, most points could be seen in a day from a point approximately on the earth's orbital plane. For the points closest to the poles, the tilt of the earth's axis would reveal them over the course of a year.
I think the answer is:
It is almost always possible if we consider the Earth a perfect sphere. There are some edge cases which can cause that there is no such angle though.
If viewed from an ideally far distance, all points on one half of the sphere (not including the bordering orthodrome) can be seen at the same time.
So the puzzle is equivalent to show a half of the sphere which contains at least 4 of the 5 points.
For example if all the 5 points are on a same orthodrome, then this might be impossible - think about 5 points which divides it in 5 equidistant pieces. In this case the problem reduces to finding a half of a circle with 5 marked points, which contains 4 of those points.
Another impossible configuration is like the one given by Gamow.
Or generalizing Paul Evans' idea, even if you have 6 points which are pairwise opposite, you can see at most one from each pair.
However, generally you can find 2 marked points, which are not on opposing poles of the sphere, and that the orthodrome defined by them does not contain any of the other 3 marked points. This orthodrome halves the sphere into two equal parts. By pigeonhole principle, at least 2 of these points will be on the same half of the sphere. These 2 and the 2 on the orthodrome can be seen from a well chosen angle.
Here's a solution similar to Gamow's but with a different arrangement and an illustration.
Building on Rodolvertice's answer:
One can simply pick any satellite dish. Together, they give coverage of the whole Earth. This makes the condition on numbers redundant however.
Alternatively, building on top of this as well
Simply use satellite images in the comfort of your own home. However, the question requires "outerspace". So choose a space station. And request access to the specified coordinates.