If only one person claims X then it might be fraud. If large numbers of seemingly unrelated people all claim X then you're forced to decide between X and a global conspiracy to misrepresent X.
To your example. Importantly, even if you deemed one of the global mean temperature datasets to be untrustworthy there are other related (but different) datasets. There are also other pieces of evidence related to the downstream claims that don't look directly at temperature.
There are many things that cannot be feasibly verified empirically without access to rare resources.