Often called the “Garden of,” the Greek word Γεθσημανί is a transliteration of גַּת שְׁמָנֵי, meaning “oil press.” The location is on the Mount of Olives. Presumably, Gethsemane was an olive orchard owned by a supporter of Jesus who allowed Jesus and his followers to stay there rather than return to Bethany. John 18:1 calls it a garden (κῆπος, a word which can refer to an enclosure), and the disciples “enter” it, implying it had a short wall marking out the boundaries of this particular orchard.
There are at least four possible sites; the most popular is the Church of All-Nations (constructed in 1919-1924, maintained by the Franciscans). The church has a small olive garden, and inside the church is the traditional “agony stone.” Early visitors reported a church at the location in the late fourth century. There was a crusader church until 1345 (Schnabel, Jesus in Jerusalem, 116).
In 2020, archaeologists discovered a Jewish mikveh while excavating a tunnel under the byzantine church. A mikveh is a pool used for ceremonial washing in the Second Temple period. This is the first archaeological evidence from the Second Temple period at Gethsemane. In the first century, Jewish workers producing wine or olive oil needed to be ceremonially clean, so if there was an olive press there, it is not surprising there was a mikveh.

Church of All Nations
The primary competing traditional location of the betrayal of Jesus is the Grotto of the Agony, near the Tomb of Mary on the Mount of Olives. The Book of John about the Dormition of Mary (a sermon written by John, Archbishop of Thessalonica in the seventh century) implies her tomb was in Gethsemane. Franciscans purchased the cave in the seventeenth century, but the Eastern Orthodox Christians continued to use the cave until 1919. The site is today maintained by the Franciscans. The cave may have had an olive press (although nothing remains). The cave would be large enough for Jesus and some disciples to get out of the weather (33 feet deep, 62 feet wide).
Joan Taylor, “The Garden of Gethsemane Not the Place of Jesus’ Arrest” BAR 21 (1995): 26–35. Taylor argues, “Jesus was arrested not at the traditional Garden of Gethsemane adjacent to the Church of All Nations but rather in the cave of Gethsemane (on the way to the Tomb of the Virgin Mary).”

Image Credit SPQR10 (derivative version). – Derivative work from the 1914 edition of the Catholic Encyclopedia, a publication now in the Public Domain., Public Domain, https://commons.wikimedia.org/w/index.php?curid=94596868
It is best to agree with Davies and Allision, “Traditional and modern proposals (e.g., the Grotto of the Agony) are guesses” (Matthew, 3:494).
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