Can units leave the Baron Pit battlefield?

Yes, units can leave the Baron Pit battlefield.

While Baron Nashor's ability requires him to enter the Baron Pit when he is played, there is no rule or card text that restricts him—or any other unit—to that location once they are on the board. Units can leave the Baron Pit just as they would any other battlefield, provided an effect or standard movement action allows it.

Additionally, the Baron Pit battlefield itself has the ability: "Units can move here from anywhere." While this facilitates movement to the Baron Pit, it does not restrict units currently at the Baron Pit from leaving it (RiftJudge FAQ #10118).