Wales have not reached a major tournament since they qualified for the World Cup finals in 1958.
The following years have been a tale of near-misses as great players such as Ryan Giggs, Mark Hughes and Ian Rush have tried and failed to steer Wales through a series of ultimately disappointing qualifying competitions.
The expansion of the European Championship to 24 teams (up from 16) has given Bale and an ever-improving Wales side perhaps their best opportunity to compete among Europe's elite.
"It would be right up there with it (Champions League) and hopefully we can do it," Bale told the BBC.
Wales have been drawn in a favourable qualifying group that includes Bosnia, Belgium, Israel, Cyprus and Andorra, with a top-three finish giving them a chance of reaching the finals.
The top two in each group qualify automatically along with the best third-place finisher, while the remaining eight third-place teams go into a playoff.
Bale, who became the world's most expensive player when he joined Real Madrid from Tottenham Hotspur a year ago for some 100 million euros ($130 million), believes this European Championship offers Wales a great chance to end the long wait.
"It's the ultimate honour for me to play for Wales," he added.
"For me I want to try my best. I want to try and make history with the team and qualify for a major tournament and that's something we are all aiming for.
"We firmly believe that we are all pushing in the right direction."
Wales face back-to-back qualifiers against Bosnia on Friday and Cyprus on Oct. 13.
(Reporting by Toby Davis; editing by Ed Osmond)
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