This is the 4582nd most frequent Russian word.
государственную
"State-owned" or "public" (relating to the state/government).
Here, 'государственную' is an adjective modifying the noun 'школу' to indicate that the school is state-run or government-funded.
Я учусь в государственную школу.
I attend a government school.
Here, 'государственную' modifies the noun 'работу,' suggesting the job is associated with or funded by the government.
Он устроился на государственную работу.
He got a government job.
Here, 'государственная' (in the nominative case, unlike the other examples in accusative) modifies 'галерея' to indicate that the art gallery is managed or owned by the government.