In video filmed by Fossil Free London, performers immediately left the stage and security confronted the group.
A safety curtain came down drawing a mixture of cheers and boos from the audience.
Noise from the crowd drowned out protesters who spent several minutes chanting at the theatre on Rosebery Avenue, Islington.
Additional eco-campaigners could be seen in the wings holding cardboard signs - one of which demanded an end to money linked to oil in arts and culture organisations.
One person on X, formerly Twitter, said the disruption lasted for around 20 minutes.
BREAKING: We’ve just disrupted a performance of Romeo and Juliet at Sadler’s Wells theatre, telling them to drop @Barclays dirty oil sponsorship! It’s time to get #OilyMoneyOut of our arts, our city and our futures 🚫 pic.twitter.com/NU9hQjUpou
— Fossil Free London (@fossilfreeLDN) August 31, 2023
A spokesman for Sadler’s Wells said: “Protesters were promptly ejected and the performance successfully restarted to audience cheers.
“We respect the right of individuals to air views on these matters, but a disruption to dedicated musicians and young dancers at an eagerly awaited performance during this sell-out run of Romeo and Juliet is wholly unacceptable, and we are supporting police to investigate the matter.”
Joanna Warrington, of Fossil Free London, said: “Like doomed star-crossed lovers, Barclays and Sadler’s Wells are trapped in a deadly romance as they exchange money for social acceptance, except in this version, after the worst effects of climate breakdown ricochet across the world, we all die.
“By taking their dirty money, Sadler’s Wells is helping Barclays to launder its dirty image and is complicit in their climate-wrecking investments. It’s time to cut oily money out of the arts.”
The group claims Barclays is investing in new oil and gas projects, saying the bank has spent more than $190bn (£150bn) on fossil fuels since the Paris Climate Accord was signed in 2016. Barclays said it is working with companies to “transition to a low-carbon business model.”
It comes after several months of action by Just Stop Oil activists who have marched in main London roads, disrupted events including The Open, Wimbledon, the Ashes, the Gallagher Premiership rugby final at Twickenham and the World Snooker Championship.
They also interrupted Chelsea Flower Show, the London Pride March and sprayed orange paint on the Department of Energy Security and Net Zero building in central London.
Barclays said: “We believe that Barclays can make the greatest difference as a bank by working with customers and clients as they transition to a low-carbon business model, focussing on facilitating the finance needed to change business practices and scale new green technologies.
“This includes many oil and gas companies that are critical to the transition, and have committed significant resources and expertise to renewable energy. Where companies are unwilling to reduce their emissions consistent with internationally accepted pathways, they may find it difficult to access financing, including from Barclays.
“As one of the first banks to set an ambition to become net zero by 2050, we are clear that addressing climate change is an urgent and complex challenge.”