The mishna in Beitza[1] says that one is not allowed to swim across a body of water on Shabbat and Yom Tov. The gemara in Shabbat[2] explains that this prohibition even includes a pool of water that is located in an enclosed area, if the pool doesn’t have a rim around it.[3] So paskens the Shulchan Aruch.[4] A few queries are posed by the poskim: whether the leniency (i.e., having a rim) applies to a pool only in a private area or even a public one, what constitutes a rim, etc. These may impact the status of an average swimming pool.
If one’s body becomes wet, he is allowed to dry himself with a towel.[5] However, he should be careful not to squeeze the water out of his hair or out of the towel, as squeezing a liquid from a solid is prohibited on Shabbat. It is recommended to let the water on one’s body drip off for a few moments so that there will be less chance of squeezing.
If one is in a place where he is not allowed to carry, he must dry himself before walking four amot[6] on the shore or shallow water, even though “carrying” the water on his body is not the regular way of carrying things.[7] This stringency does not apply while moving around in the body of water or when in the rain. Thus, in a case where swimming is permitted, one would not need an eiruv while in the water. However, outside the water, it’s somewhat difficult to manage without an eiruv, unless one dries himself off and leaves the towel right next to the pool or waits a while to drip dry. (Regarding your reference to an eiruv, you probably realize that an eiruv is just one way to have a reshut hayachid (private domain), where carrying is permitted. Enclosed areas are similarly a reshut hayachid and, in many cases, are places where it is permitted to carry. If, however, the enclosed area is joint to more than one living unit, it may require a different type of eiruv. However, that matter is well beyond our scope.
The above is the halachic background for your question. However, halacha l’ma’aseh, the poskim note that the accepted practice is not only to forbid swimming, but not to allow bathing either, even in cold water and even if the conditions referred to above are satisfied.[8] The Magen Avraham[9] states several reasons for our minhag, including the concern that one might squeeze out some water while drying off.