In the Majority Opinion excerpt, they state: "We uphold the exclusion order. …. Compulsory exclusion of large groups of citizens from their homes … is inconsistent with our basic governmental institutions." Which government institutions are they referring to?

The Government Institution they are referring to is the Government Defense Institution/ Millitary which is  aimed to protect its citizens.

Explanation:In 1942, Franklin Roosevelt signed Executive Order 9066 which made the US military to make part of the   United States a military zone which prompted a compulsory exclusion of  large  groups of people from their homes which were mainly the  Japanese-Americans who were taken from their homes and placed in internment camps during World War II.

Frank Korematsu, a  Japanese- American defied the order to be relocated and be reported in the camp which led to his arrest and conviction. The Supreme Court ruled against him on the basis that  the internment of Japanese- Americans during wartime  was constitutional.

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explanation:

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26. having important positions chosen by lot