a beam: the heavy wooden timber that carries the roof of a house
This is a builder's word. In everyday Greek it named the main timber of a house, the thick beam laid across the walls that the lighter roof pieces rested on, and it could also name the heavy bar that held a gate shut. Building accounts and a contract for an oil press use it exactly that way, and one ancient historian described a Roman battering ram as a beam "as big as a ship's mast."
The Greek Old Testament keeps it just as concrete: the roof over Lot's house, the cedar beams of a house in the Song of Songs, the trunk the prophets felled by the Jordan that took an iron axe to cut. The word never shrank to "stick" or "splinter," and it never turned into a figure of speech for a moral fault. Alongside it, ξύλον is the general word for wood or anything wooden; this one is a specific worked timber with a job to do.
All five uses in the New Testament are one saying of Jesus, told in Matthew and again in Luke: a man has a beam in his own eye and offers to take a speck (κάρφος) out of someone else's. Both objects come from the same workshop, one the smallest scrap of dry matter, the other the largest piece in the frame of a house. The word carries its full weight there, so the picture is not an exaggeration that can be toned down. It is deliberately impossible, and the impossibility is the lesson.