Workers called the 24-hour strike because they fear that some of country's landmarks will be included in a fund of assets to be privatized, something the government has denied.
Archaeological sites and several museums were shut in protest including the Acropolis hill and the Parthenon, the ruins of an ancient Greek temple to the goddess Athena on top of the hill that overlooks the capital.
"We are disappointed. We are here only for one day and we will not be able to visit the Acropolis," Alexander Walter, a German civil servant on a cruise, said a few meters from its gates, where protesters hung a banner reading "NOT FOR SALE."
A few dozen workers held up banners reading: "The archaeological culture of the country is not for sale."
The workers are demanding the government announces which cultural sites are being considered for the fund and what sale would mean.
"We don't know which monuments and which museums are on the list," said the head of their union, Stamatia Marketou.
Greece's finance ministry on Wednesday said archaeological and historical sites, monuments and museum are exempt from the fund of assets to be privatized or under private management.
The culture ministry too has denied it.
"What's happening here today is out of place, to say the least," Culture Minister Myrsini Zorba told state TV.
Greece, which is emerging from years of economic crisis, has had record tourism this year, with more than 30 million visitors.
(Reporting by Lefteris Papadimas; Writing by Karolina Tagaris; Editing by Matthew Mpoke Bigg)