Can a circle truly exist? Is a circle more impossible than any other geometrical shape? Is a circle is just an infinitely-sided equilateral parallelogram? Wikipedia says...
A circle is a simple shape of Euclidean geometry consisting of the set of points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius.
A geometric plane would need to have an infinite number of points in order to represent a circle, whereas, say, a square could actually be represented with a finite number of points, in which case any geometric calculations involving circles would involve similarly infinitely precise numbers(pi, for example).
So when someone speaks of a circle as something other than a theory, are they really talking about a [ really big number ]-sided equilateral parallelogram? Or is there some way that they fit an infinite number of points on their geometric plane?
 A: A. B. Kempe's lecture on linkages How to Draw a Straight Line (1866/1867) suggested that a circle was a much easier construction than a straight line.

As regards the circle we encounter no
  difficulty....The apparatus I have just described is
  of course nothing but a
  simple form of a pair of compasses,
  and it is usual to say that the third
  Postulate postulates the compasses.
But the straight line, how are we
  going to describe that? Euclid defines
  it as “lying evenly between its
  extreme points.” This does not help us
  much. Our text-books say that the
  first and second Postulates postulate
  a ruler (2). But surely that is
  begging the question. If we are to
  draw a straight line with a ruler, the
  ruler must itself have a straight
  edge; and how are we going to make the
  edge straight? We come back to our
  starting-point.

A: In the same sense as you think a circle is impossible, a square with truly perfect sides can never exist because the lines would have to have infinitesimal width, and we can never measure a perfect right angle, etc. 
You say that you think a square is physically possible to represent with 4 points, though. In this case, a circle is possible - you only need one point and a defined length. Then all the points of that length from the initial point define the circle, whether we can accurately delineate them or not. In fact, in this sense, I think a circle is more naturally and precisely defined than a given polygon.
A: It may be interesting to note that the nature of the circle on the basis of the Euclidean axioms is somewhat less than one might think, and is consistent with some strange behavior. The reason is that Euclid had no formal continuity assumptions as a part of his axiomatic framework, and it turns out that the rational plane $\mathbb{Q}\times\mathbb{Q}$, consisting of points having only rational coordinates, satisfies all the Euclidean axioms. 
This fact can be used to show that some of Euclid's arguments and constructions are not actually correct. For example, Euclid describes how to construct the perpendicular bisector of a line segment KM, by constructing circles P and R with radius KM and joining the intersection points A and Z as below. The line AZ is the perpendicular bisector of KM.

But the difficulty here is that Euclid never proved that circles with a common radius must intersect, and so he doesn't know that A and Z actually exist. The fact that A and Z exist is a hidden unstated continuity hypothesis in the system. Worse, it is consistent with Euclid's axioms that the circles do not actually intersect, in that there are no such points A and Z. For example, this is the case in the rational plane $\mathbb{Q}\times\mathbb{Q}$, since when K and M are rational, then A and Z are not. And so it is not possible to prove on the basis of Euclid's axioms that the circles do intersect in points A and Z. The circles may simply somehow pass through each other without touching. In the rational plane, these circles do not intersect, but instead pass through each other without meeting, and this construction does not succeed in building the perpendicular bisector.
The conclusion is that it is entirely consistent with Euclid's axioms that circles have these strange holes in them and that circles with a common radius may not intersect. 
There are several other similar issues with the Euclidean axioms, and these led to various formal corrections to and axiomatizations of the Euclidean axioms in the early twentieth century. 
A: I think it quite depends on how you understand "exist" and how you define $circle$. For example, in terms of Cartesian coordinates, a circle of radius $1$, centering at the origin of the $x$-$y$ plane, can be "defined" as the set 
$$\{(x,y)\in{\mathbb R}^2:x^2+y^2=1\}$$
Then when you ask whether such set exists or not, the answer is yes, according to the axioms for set theory.
The keyword in the excerpt you quote from wiki, I think, is "set"(the set of points in a plane that...). So when you are asking "can a circle exist" you are actually asking "can such set exist".
A: Your question depends on the nature of "existence." There are two possibilities, as I see it: 1. The world is fully mechanical and made up of a finite set of measurable parts; or 2. The world is infinite and has no smallest measurement.
This is important due to the irrationality of a circle's measurements. You can never have a circle with a rational circumference. So, it would be impossible for the circle to be made up of indivisible parts. The last part will always be slightly offset. However, if it is not necessary to consider the construction of the circle based on parts, then yes it would be possible. 
I'd say that a circle is possible when referring to space, but not when referring to matter.
