At the age of 5 he moved to Kyiv. He started playing football at the age of 6, when his father brought Yevhen to enroll in the "Dynamo" school in Nyvky. The first coaches of the football player were Vitalii Khmelnytskyi and Yurii Yastrebynskyi, but the players did not have a special position then. Yevhen Rudakov scored for "Volynets".
In CYFLU, "Volynets" started playing for FC "Vidradny". In the 2006/07 season, he played 23 games (all in the starting lineup), winning bronze medals in all-Ukrainian competitions, and the following season he played for "Dynamo" in the 15-year-old tournament.
He played his first match for the "white and blues" on April 14, 2008 in Chernihiv against "Yunist", replacing Oleksandr Tkachenko at the beginning of the second half. "Dynamo" of this age group has a whole galaxy of talented goalkeepers - in addition to Volynets and Tkachenko, the coaches also entrusted a place in the team to Myroslav Bonya.
In that season, "Dynamo" U-15 became the champions of Ukraine, beating "Shakhtar" Donetsk in a decisive match against their main rival. Volynets played a full match, keeping his goal in the castle. In total, in the 2007/08 season, Yevhen went on the field in six games, but the following season he had 14 matches, 12 of which were in the starting lineup.
In the 2009/10 season, the "Dynamo" team, played by Volynets, again became the best in Ukraine. Once again, the "white and blues" met "Shakhtar" Donetsk in the decisive match. No goals were scored in the main period, and Volynets became the protagonist of the post-match penalty shootout. After repelling two blows from "Shakhtar", the goalkeeper allowed "Dynamo" to take the lead with a score of 3: 1, but his partners did not make the next two attempts to put an end to this lottery. "Shakhtar" was the first to strike an additional sixth blow, and Volynets again appeared at the height, who was later recognized as the best goalkeeper of the final tournament.
In the 2009/10 season, Volynets, even before the completion of the "Dynamo" Academy, was included in the application of "Dynamo-2" for several matches of the first league. In the 2010/2011 season, Volynets has already been included in the first league games 13 times. Then the number of inclusions in the application for the match was reduced to 5, and finally, in the 2012/13 season, he was again on the bench 13 times. Despite this, he never went on the field for four seasons.
Volynets made his debut in the official match on October 23, 2013, where he played a full match of the first league against Kharkiv's "Helios" and kept his goal "dry" (3: 0). Until the end of the season he played in 14 matches of the championship, and in the following he continued to be involved in the matches of the second "Dynamo" team in the First League.
In July 2016, he became a player of Kharkiv's "Helios".
In 2018, he became a player of FC "Kolos", with whom he joined the UPL, where he became the team's main goalkeeper and helped the team to qualify for the Europa League after their debut season.