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The Bavli, Shabas 31 amud 1, has two stories in which a gentile came to Shamay asking to be converted to Judaism, upon which Shamay "pushed him with the construction cubit in his hand". Rashi explains that a construction cubit was a cubit-long stick used in construction for measurement.
Why did Shamay have one in his hand?
I heard once in a recording from R. Y.S. Schorr that Shamai represented a middas hadin, an exacting attitude of strict justice (as is evidenced by those very stories). His measuring stick was a display of just that point - everything had to be measured and exactly according to what was deserved.
The Chassam Sofer seen here s.v. על רגל אחד says that Shammai besides being a Talmid Chacham was also a builder by trade, כי הוא היה תלמיד חכם וגם בנאי באמת הבנין.
The apikores thought that a person must choose between being a good person who interacts with other people kindly, or one can dedicate their life to Hashem as a religious recluse. Shammai wanted to show him that it is actually possible to serve Hashem in both regards, and proved this by chasing him with the tool of his trade and allowing the apikores to recognize he was both a talmid chacham and a builder (i.e. involved in a trade and interacting kindly with people).
Hillel's response was to say that you can 'learn on one foot', by which he meant to only involve yourself in kindness. However explains the Chassam Soffer, don't forget that the single foot of kindness includes kindness to Hashem, which means keeping His torah.
The Ben Yehoyada (in one explanation here), similarly, says he was a building-measurer by trade, and so carried a stick with him.
in chovos halevavos gate 9 ch.5
Do so in your heart and mind if you are unable to free your body to matters of the next world due to being so fully engaged with providing for your livelihood and maintenance, as our Rabbis mentioned on many (great men), who would toil in matters of this world, while being separate from it (in heart and mind), such as Aba Chilkiya who would work hoeing the ground (Taanis 23a), Shammai in construction work, and Hillel who would earn his livelihood by chopping trees and selling the wood (Yoma 35b).
The Maharsha there s.v. על רגל אחת explains that Shamai was, in the two accounts in which he was holding the construction cubit (as he was not reported to be holding it in the first account), showing the questioner some important element of his response.
The fellow who asked to be taught the entire Torah on one foot was being shown that just as a building cannot stand on one pillar, so too the Torah has many mitzvos which cannot be given one simple foundation.
The fellow who asked to be a kohen gadol was being shown that the Torah does not stand on the pillar of kehuna (priesthood) alone.
