First, we should understand that the laws of an eiruv techumin, the halachic device that allows one to walk where he otherwise could not, are different from those of an eiruv chatzeirot (the latter is the “eiruv” of common parlance), which allows one to carry within a city. We will summarize the basics, based primarily on Shulchan Aruch, Orach Chayim 396, 398, and 408.
The Torah forbids one to “leave his place on the seventh day.”[2] A person’s place is defined as the area of a city, which, regarding the laws of Shabbat, is derived from the Israelite encampment in the desert, namely, 24,000 amot by 24,000 amot. In addition, the Rabbis forbade walking more than 2,000 amot in any direction from his place of habitation or base.[3]
Let us investigate what a person’s base is. Every person’s minimum base consists of a radius of four amot around him.[4] However, if a person is in an area that is fully enclosed for the purpose of human habitation (including by an eiruv chatzeirot[5]), that whole area is his base; beyond that is the techum (the 2000 amot boundary zone).[6] Even if an area is not enclosed, an uninterrupted cluster of human habitation is considered the base for those who commence Shabbat within it. They may walk throughout the cluster plus an additional 2,000 amot beyond its extremities. From a geometric/halachic perspective, it is difficult to determine whether some region is an uninterrupted area and to map out its boundary. The local rabbi(s) should make these determinations after studying the region’s layout.
We will mention some rules after pointing out that conventional halachic wisdom is that within built-up, residential areas of cities, one can usually walk to wherever he has occasion, even beyond the four-kilometer distance you mentioned. (Rural areas are much more complicated.) Homes that are within 70⅔ amot of a cluster of habitation count as its extension. If the 70 2/3 amma radii around these residential clusters overlap (i.e., the clusters are less than 223 ft. or 68 m. apart at the shortest distance), they are considered a single cluster.[7] After determining the city’s boundary, one encloses it (assuming the boundary is jagged) with a north-south/east-west rectangle.[8] This usually increases the block’s size significantly (and, according to some, connects it to other areas). From that point, an “outer box” is “drawn” 2,000 amot around the rectangle. Even if the outer rectangle extends into a new cluster, one may not walk past its boundaries.[9]
An eiruv techumin does not increase the distance one is allowed to walk. Rather, it alters a person’s base for Shabbat, from which we measure 2,000 amot in all directions and within which he may walk. This can be done either by being physically present at the desired base as Shabbat begins or by placing food there before Shabbat and properly proclaiming his intent to make it his base.[10] The latter is the eiruv techumin.
The advantage of the eiruv techumin is that it allows one to travel further than 2,000 amot in one direction. For example, if he puts the eiruv techumin 1,500 amot west of his home, he may walk 3,500 amot to the west on Shabbat, but only 500 amot to the east. One can have only one base during the course of a Shabbat, and therefore multiple eiruvin do not work.
If one puts an eiruv techumin in a place where he is halachically unable to go according to the rules of techumin, it is ineffective.[11] However, the laws of clusters of homes can be helpful. If you put an eiruv techumin somewhere within a cluster that extends to within 2,000 amot of your location on one side and 2,000 amot of the synagogue on another, you may walk more than 4,000 amot, as the entire central cluster, no matter how large it is, constitutes your base and does not count toward the 2,000 amot of the techum.[12]
Your local rabbi should be able to tell you whether an eiruv is needed and will work in your situation. If it is required, he will show you where and how to place it.