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The LIGO complex is very large, but I assume that there is a very small chamber that actually does the science since they are measuring very small sheers. How big around is the actual evacuated chamber?

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The arms are $4\,\mathrm{km}\,\times\, 1.2\,\mathrm{m}$:

From the LIGO webpage:

The 1.2 m diameter beam tubes were created in 19-20 m-long segments, rolled into a tube with a continuous spiral weld. While a mathematically perfect cylinder will not collapse under pressure, any small imperfection in a real tube would allow it to buckle (a crushed vacuum tube would be catastrophic). To prevent collapse, LIGO's tubes are supported with stiffener rings that provide a significant layer of resistance to buckling under the extreme pressure of the atmosphere. The tubes must withstand these stresses for at least 20 years.

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LIGO detectors have two 4 km arms, and everything in those arms in suspended in a vacuum. The actual dimensions of those arms I was unable to find, but the published paper here has more information for you to peruse.

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