What if the evidence is in a form they cannot take, or goes against something long-established?
What do you mean by "in a form they cannot take"?
Do you mean something like a person's claim based upon a vision they claim to have had?
If so, that isn't evidence.
As for going against something long established, do you mean like how Einsteinian relativity went against Newtonian/Gallian relativity?
Or how wave particle duality went against the very nature of what it meant to be a particle or a wave?
Or how light propagating through a vacuum went against the firmly established idea that all waves need a medium to propagate which lead some to cling to aether with many contradictory properties?
Or for more ancient examples, the evidence showing Earth was round, going against the established belief that Earth was flat; or the evidence that Earth was moving going against the established belief that Earth was the fixed centre of the universe?
So it has happened quite a bit in science.
It is a fundamental part of science.
You can believe yourself open to any fact, but it does not mean anything if you will only accept things that can be shown through limited means.
That depends upon what those limitations are.
If those limitations are just so that things can actually be shown, rather than just asserted, then that is still being open.
Not accepting unsubstantiated claims does not make someone close minded, no matter how much other people believe those claims.
It is when you start to dismiss evidence because it doesn't agree with your beliefs that you are close minded.
There are whole fields of discovery beyond the experimental that countless people dedicate their lives to.
Do you mean like the people in many contradictory religions that all think their religion is the one true religion and all the others are wrong?
If we actually accepted that as a valid way of determining what is true, then multiple contradictory religions would be accepted as true and your beliefs would contradict themselves.
Scientists use experimental evidence because it is what has been reliable, what can actually be used to find out the truth, even if it isn't the complete truth.
If you can provide an alternative which actually works, feel free to do so. Until then, I will continue with what works.