The entrance to the museum has been closed to visitors while the demonstrators target the Our Future Planet exhibition.
Eleven protesters, among them scientists dressed in lab coats, have glued or locked themselves to railings inside the museum.
Supporters have been outside the London landmark during the afternoon, waving flags and banners, and chanting: “No more petrol, no more diesel, funding fossil fuels is evil.”
A 12ft model of a pink dodo - the bird that was driven to extinction in the 1660s - was erected by protesters.
Extinction Rebellion has criticised the Science Museum for taking funding from Shell for the Our Future Planet exhibition, which began on May 19 and runs until September.
Teenage climate activist Greta Thunberg has also hit out at the Shell sponsorship after previous reports said the museum had signed a gagging clause over the exhibition funding.
The exhibition explores the technologies being developed to remove carbon dioxide from the atmosphere.
Dr Charlie Gardner, an associate senior lecturer in conservation science at the University of Kent, and a member of Scientists for Extinction Rebellion, told the PA news agency: “Shell, an oil company, is a major driver of the climate emergency.
“We find it unacceptable that a scientific institution, a great cultural institution such as the Science Museum, should be taking money, dirty money, from an oil company.
“Oil companies have invested heavily over the last three decades in undermining public confidence in science and it’s the reason why there are still, to this day, people that doubt or deny the existence of climate change.
“Oil companies have an anti-science agenda and yet the Science Museum is taking their money.
“The fact that Shell are able to sponsor this exhibition allows them to paint themselves as part of the solution to climate change, whereas they are, of course, at the heart of the problem.”
Hayley Pinto, an addiction psychiatrist from Norfolk, and a member of a group of doctors supporting Extinction Rebellion, told PA: “The Science Museum is somewhere I used to bring my kids when they were little.
“Meanwhile we’ve got Shell, a company who’s investing only a tiny percentage - far less than they said would - of their money in renewable energy whilst getting on with business as usual.
“It’s destroying the future for our children. The impact on children already in terms of their health and climate change is huge.”
A Shell spokesperson said: “Shell and the Science Museum have a long-standing relationship based on a shared interest in promoting engagement in science, which will be a key enabler in addressing the challenge to provide more and cleaner energy solutions.
“At Shell our target is to become a net-zero emissions energy business by 2050, in step with society.
“As Shell works with our customers to identify the best paths to decarbonisation, we seek to avoid, reduce and only then mitigate any remaining emissions.
“Developing carbon capture and storage and using natural sinks are two of a range of ways of decarbonising energy.”
A spokesperson for the museum said: “Throughout the day our team has facilitated a peaceful protest at the Science Museum, ensuring the health and safety of everyone in the building while enabling thousands of people to enjoy their visit to the museum.”