[3] After having played lower league football in the Jutland Series for FC Skanderborg and Tjørring IF, Baun moved to Skive IK in the second-tier Danish 1st Division in January 2017.
[4] On 16 September 2017, it was announced that Baun had signed a three-and-a-half-year contract with Hobro IK effective from 1 January.
[6] Baun made his professional debut on 28 February 2018 in a Danish Superliga game against FC Copenhagen.
Coming on as a substitute for Edgar Babayan in the 77th minute, Hobro lost 0–2 after goals from Viktor Fischer and Rasmus Falk.
[9] As Hobro suffered relegation to the Danish 1st Division in the 2019–20 season, Baun would see his playing time increase.