Coach Oleg Blokhin, who took over last April after two appointments that proved less than successful, said he has no illusions about their prospects.
"We have been set a task of winning the tournament," he told Reuters in an interview. "I have no idea where that comes from. Of course, you have to aim for that, but you also have to take a sober view of things.
"First of all, we have to qualify from the group."
Blokhin, a legendary striker for the old Soviet national team, took Ukraine to the World Cup quarter-finals in 2006 and is the most successful coach in Ukrainian history.
But, though he is a strong motivator, Blokhin, 59, will have to work hard to weld a Ukrainian side of young talent and some experience into a team capable of holding their own against France, England and Sweden.
Recent form exposed a weak and unbalanced defence and an attack in which most of the goalscoring has come from midfielders, such as Andriy Yarmolenko, Oleh Gusev and Yevhen Konoplyanka, rather than the strikers.
On top of those problems, Ukraine's first choice goalkeeper Oleksandr Shovkovsky has been ruled out of the Euros with a shoulder injury picked up in a domestic match at the end of April that will keep him sidelined for at least three months.
The 37-year-old's absence will be a major blow for the side with Ukraine going into the tournament with a shortage of experienced keepers after Oleksandr Rybka was suspended for using a banned diuretic and Andriy Dykan is also out injured.
Ukraine have been beaten easily by Brazil, Italy, France, Uruguay and Czech Republic in their latest friendlies - though they are hoping a 3-3 draw with Germany might mark a turning point.
"We should have refreshed the team after the World Cup 2006 but it proved hard to do," said Blokhin.
"But you have got to have faith in yourself. We will never make it if we regard ourselves as underdogs to France and England."
Blokhin was brought after a fiasco involving 61-year-old Myron Markevic, who had originally been entrusted with shaping a team worthy of Ukraine's ambitions.
Markevic, who combined his job at the national team with coaching Metalist Kharkiv, quit after the Ukraine FA accused his club of fixing a match in 2008.
Despite pulling off three wins and one draw against the Dutch team during a three-month spell, Markevic chose to stick with Metalist in protest against the federation's ruling.
Yuri Kalitvintsev then stepped in as caretaker, but poor form in the friendlies doomed his tenure and national selectors turned again to the evergreen Blokhin.
"I have had very little time. It would have been easier if I had taken the team over earlier. I would have had more time to come up with the decisions I need to make now", Blokhin said.
Ukraine's first European championship will also be a last chance to sparkle for 35-year-old icon Andriy Shevchenko.
Team captain and all-time top-scorer, Shevchenko has been struggling with back problems since he left Chelsea in the hope of making an impact in Euro 2012.
"Everything is down to him. Names do not play football. There is a certain amount of work to be done and Andriy has problems with his health," Blokhin said.
Euro 2012 will also be a swansong for the mature and forceful 32-year-old striker Andriy Voronin who has had a fantastic season at Dynamo Moscow, but often finds it difficult to reproduce his club form for his country.
Blokhin frankly admits he is concerned over the lack of attacking power.
"I am pleased the midfielders are scoring a lot but we lack a striker who will keep the opponents' defence in tension", he says.
Bayern Munich's Anatoliy Tymoschuk, their most capped player, will inspire his partners in midfield alongside tried-and-tested Serhiy Nazarenko and Oleh Gusev.
At the same time, Euro 2012 will be a chance for 22-year-old attacking wingers Yevhen Konoplyanka and Yarmolenko who have now found their places in the main squad, both scoring in friendly games against Uruguay and Germany.
The unbalanced defence is a concern. Blokhin has not had a clear chance to assemble his key men since he was re-appointed.
Injury kept influential central backs Taras Mykhalyk and Dmytro Chygrynskiy on the sidelines last season. Chygrynskiy has still to play his first game under Blokhin's leadership and is not in the squad for the finals.
In a country prone to superstition, Ukraine have something else to be concerned about as well.
They have been drawn to play against England and France in the eastern city of Donetsk. This has been a luckless venue for them and Ukraine have still to win their first game at Donetsk's highly impressive Donbass Arena.