Time flows slowly on the island of Osaki-Shimojima, part of an archipelago off the southern coast of Japan’s main island, Honshu. Elderly fishermen pass the days on its stone embankments, trying to catch fish in the Seto Inland Sea. Some of the wooden buildings have stood along the narrow alleys of the island’s historical district of Mitarai for 250 years; most have roofs of the same traditional clay tiles, called kawara.

One of those buildings houses the Shinko Clock Shop, widely believed to be among the oldest clock and watch shops in Japan — although no one, not even the owners, knows precisely when it began to sell timepieces

“We cannot confirm accurately whether it’s the oldest, but it’s likely one of the oldest,” said Mitsushi Matsuura, 45, the fifth generation of his family to sell clocks.

Based on the dates on a local grave marker and some family documents, the Matsuuras say they believe the shop was established in the late 1700s to sell rice and other foodstuffs, then started to sell clocks sometime during the Meiji era (1868-1912).

Now, Mr. Matsuura and his father, Keiichi, 79, handle sales and repairs, with family members taking on the accounting and other tasks.