Kirkjubaejarklaustur in Sida was known in olden times as "Kirkjubaer" (Church Farm) and was an important farming estate. Kirkjubaejarklaustur has developed into a village, the only centre of population in the district, with about 150 inhabitants. Kirkjubaejarklaustur, often abbreviated to "Klaustur", is centrally located in the district. Roads radiate from Kirkjubaejarklaustur in many different directions: the Ring Road (No. 1) runs through the district. The Laki road, just west of Kirkjubaejarklaustur, leads into the highlands. The circular Landbrot / Medalland road serves the southern part of the district. The Fjallabak roads (north and south) lead from the Ring Road into the interior via Skaftartunga. The Álftaver road is a circular route serving Alftaver on Myrdalssandur.

Kirkjubaejarklaustur has a long and interesting history. Irish hermits, "Papar", are believed to have lived at Kirkjubaer before the Norse settlement of Iceland. Tradition says that it has always been inhabited by Christians, and that pagans were unwelcome. The 9th-century settler Ketill the Foolish made his home at Kirkjubaer. After Ketill's time, Hildir Eysteinsson from Medalland, a pagan, attempted to move to Kirkjubær. When he set foot on the estate, he fell down dead, and was buried in Hildishaugur (Hildir's Mound), a rocky hillock just east of Kirkjubaejarklaustur. In 1186, a Benedictine convent was founded at Kirkjubaer. The convent was active until disbanded at the reformation in 1550. Many local placenames and folk tales reflect the presence of the nuns and ecclesiastical history down the centuries.

Systrastapi (Sister's Rock) is a steep-sided rocky hill west of Kirkjubaejarklaustur. Folklore says that two nuns were buried up on the rock after being burned at the stake for breaking their vows; one is supposed to have sold herself to the Devil, carried consecrated Communion bread by the door of the privy, and had carnal knowledge of men. The other spoke blasphemously of the Pope. After Reformation, the second nun was regarded as innocent, beautiful flowers grew on her grave, while the guilty nun's grave remained barren.

Systravatn (Sister's Lake) is located above Kirkjubaejarklaustur, at the edge of the rocky belt. The nuns of the convent often went there to bathe. One day, two of them saw a hand emerge from the lake, wearing a fine golden ring. They seized hold of the hand, and were dragged down into the depths. From Systravatn, Systrafoss (Sister's Falls) cascades down into the Fossargil gully.

Kirkjugolfid (The Church Floor) just east of Kirkjubaejarklaustur, is a bizarre expanse of columnar basalt, eroded and shaped by wind and waves. It is a protected natural monument.

Kirkjubaejarklaustur Chapel, consecrated in 1974, was built in memory of the Rev. Jon Steingrímsson, who on 20th July 1783 celebrated the "Fire Mass" in the church at Kirkjubaejarklaustur, and was credited with halting the flow of lava which threatened the area. The site, now known as Eldmessutangi (Fire-sermon Point) is west of Systrastapi. The new chapel, which stands slightly farther to the east, was designed by architects Helgi and Vilhjalmur Hjalmarsson.