In 578 CE, the Islamic Prophet Muhammad was a pre-teen boy, the Roman Empire was a shadow of its former self with the Byzantine Empire rising from the ashes, and Shigetsu Kongō started the company Kongō Gumi (株式会社金剛組) in Osaka, Japan. The company started with a contract from Prince Shōtoku Taishi to build Shitennō-ji, the first Buddhist Temple in Japan. Shigetsu Kongō was from Baekje, a Buddhist state in what is now Korea. For over 1400 years, the company specialized in building Buddhist temples using techniques used by Shigetsu Kongō in 578. The company was run by the same family through its entire history with sons-in-law taking the Kongō family name and, in one case, the widow of the 37th leader became the leader and helped the company survive the Shōwa Depression (1927).
After surviving many periods of warfare and unrest, including anti-Buddhist periods as well as many financial crises (such as the big Shōwa Financial Crisis in 1927), Kongō Gumi had to accept becoming part of Takamatsu Kensetsu in 2006. The company lost some big bets on investment properties from which it was unable to recover on its own. Takamatsu maintains Kongō Gumi as a separate subsidiary and it still builds Buddhist temples.
Currently, the oldest business still in operation independently is a hot spring hotel in Hayakawa, Yamanashi Prefecture, Japan founded in 705 CE named Nishiyama Onsen Keiunkan (西山温泉慶雲館 ). Like Kongo Gumi, the hotel has been owned and managed by the same family for all of its years. Obviously, the physical hotel has been renovated a number of times over the past 1300 years but it still gets all of its hot water from the hot spring just as it did in 705.
You can find a list of the oldest companies at Wikipedia.