Born and raised in Sinj, Barać started his youth career at the local Junak Sinj. At the age of 17 he joined the Hajduk Split youth academy, winning the national U-19 championship with the club, but did not receive a professional contract at the end of his academy period, so he returned to third-tier Junak.[4]
Following his first senior-level season playing in his hometown, he signed a three-year contract with the Swiss second-tier team FC Wohlen in July 2014,[5] but his contract was rescinded the following month as Barać couldn't get a Swiss work permit.[6]
Barać then moved back to Croatia, signing with second-tier Hrvatski Dragovoljac for the remainder of the season, but missed most of it due to injury.[7]
NK Šibenik
Barać joined second-tier Šibenik in the summer of 2015, which was at the time coached by Mirko Labrović, his fellow native from Sinj, playing regularly and almost qualifying for the first tier.[8][9]
Osijek
On 1 July 2016, Barać joined Osijek for an undisclosed fee. Impressive display in the first half of the
Prva Liga earned him a call-up for Croatian A team for 2017 China Cup. On 3 August 2017, Osijek defeated the Dutch giants PSV Eindhoven in the third qualifying round of the 2017–18 UEFA Europa League with a 2-0 aggregate win. Many labeled Barać as the key member of the historic victory and Ajax eventually made a 2.5 million euros transfer bid. Osijek and Barać accepted, however the transfer was stopped due to Barać's heart issues revealed on the medical tests.[10]
Rapid Wien
On 12 July 2018, he joined Rapid Wien for a fee of €1,300,000.
Sochi
On 19 June 2021, he moved to Russian club Sochi as a free agent.[11]
Krylia Sovetov Samara
On 19 February 2022, he was loaned to Krylia Sovetov Samara until the end of the season with an option to buy.[12]
On 31 May 2022, Barać moved to Krylia Sovetov on a permanent basis and signed a two-year contract.[13]
Loan to Oostende
On 31 January 2023, Barać joined Oostende in Belgium on loan until the end of the season, with an option to buy.[14]
Aktobe
On 9 February 2024, Krylia Sovetov announced Barać's transfer to Aktobe in Kazakhstan.[15]