What does “modest station of judges” mean?

Solution 1:

In that context station refers to:

the position, as of persons or things, in a scale of estimation, rank, or dignity; standing:

  • the responsibility of persons of high station.

(Dictionary.com)

He is sort of downplaying the effective role and power that judges have in the American democratic system.

Solution 2:

You have the definition of station correct. He is using it to mean the post or role of a judge. By referring to it as a "modest station" is he stating his opinion that the judicial branch has a limited role in American government, and should not exercise powers beyond that in making decisions.

Solution 3:

This is the fifth sense of station in the OALD:

(old-fashioned or formal) your social position

It is in quotes because it is an exact quotation from his introductory statement.

Gorsuch is commenting that he believes the judiciary, in a democracy, should be cautious in the use of their power, as a way of assuaging the concerns of the opposition that he would use his position on the Supreme Court to overturn laws and protections they favor.

Solution 4:

What does “the modest station of judges” mean? Does it simply mean a post? What is the special implication of “station” here in quotes, if it has?

The word "station," as used here, is a formal and somewhat archaic, literary way of referring to status in society. The judge is speaking in a very formal setting, and this choice of words adds a subtle touch of gravitas and learning to what he is saying.

This meaning of "station" is associated with the phrase "station in life," and a search in Google Ngrams suggests that this phrase was most widely used in writing in the 19th century.

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Judge Gorsuch wants to reassure his audience, particularly Democrats, that he understands that a judge should not be an activist. In other words, he understands that membership of the Supreme Court is not supposed to be an opportunity to advance a political agenda to make changes in the law that will have a radical impact on American society. The word "modest" expresses this overtly, while the choice of "station" — as opposed to "position" or "status"— subtly suggests reliability, an effect that derives from its association with an old and literary phrase.