Yes, extraordinary claims require extraordinary evidence, but perhaps this is only due to our local standards of "extraordinary".

• The more extraordinary the claim, the greater the burden of proof. However, some argue that because Sagan never defined what "extraordinary" means, people have used the decree to discount immediately any idea that doesn't fit with current orthodoxy. In fact, there are plenty of novel ideas which would still require tons of proof but which, under Sagan's inadvertent orthodoxy, might unfortunately be seen as something that no evidence could ever suffice to fully prove. See examples below.

• Being that I personally think the burden of proof is always pretty much impossible to ever truly satisfy, I think that novel ideas about the world can be accepted with slightly less evidence than perhaps a misconstrued conception of this quote would call for. This is perhaps only for those ideas at the extreme ends of the spectrum, however, and at that point (think, e.g., theories of the multiverse) we may already be in unfalsifiable territory anyhow. The larger point, then, is that these ideas may read to us as being encased in impenetrable, unfalsifiable territory for now, but we may simply be unable to imagine ways in which our own imagination might evolve into something else and thereby understand or even interact with phenomenon differently. At some point, these ideas might be falsifiable (or something else we don't yet understand!), and we may miss possible ways forward now by eschewing them altogether.

• An example of an idea that seems unfalsifiable for now but which, should we start to disregard our perhaps somewhat arbitrarily drawn determination of its in fact being unfalsifiable, may have vaguely testable ways forward instead:

  • "Multiverse immortality means that death – from the observer’s point of view – is impossible. In other words, for any state of any observer with time-stamp t0, there is a possible next state t0+1, and, as everything possible exists, such an observer exists. Thus, a last moment of experience is impossible, and death – as the cessation of the process of experiencing – is impossible," writes Alexey Turchin in the Journal of Evolution and Technology on the practical implications of quantum immortality.

• Of course, there are all sorts of things going on here; for example, we are banking on the notion of "all things being possible" and "all those possible things" being in existence in some manner.

• However, an extraordinary claim like this would be something that might be written off arbitrarily by following the clipped interpretation of Sagan's quote, since doing that leaves us imagining that the concepts of "everything" and "total possibilities" we use within our theories are ultimately unbroachable for measurement or total understanding. They may be; they may not be. We might dash our chances of finding out if we put the burden of proof too high. We may, that is, never reach that larger point of expanded understanding that could come from different forms of, say, quantum technology that enable us to test some of these multiverse concepts. If our descendants do discover scientific avenues like this, it's quite possible they would forget to return to ideas of the multiverse that we have formulated in this most recent era, instead creating newer ideas about such things as the multiplicity of reality that seem to them still quite hopelessly unbroachable when, in truth, they could then be in a position to actually understand some of our ideas from now against their evidence rather than only those newer sets of ideas they formed in its face. Perhaps they will not forget about our ideas if we keep our ideas strong and open and aim to not pass them down with the mark of being completely and irrevocably falsifiable.

• As a peripheral note, a way of testing quantum immortality that does not involve uncertainty from the perspective of the observer who wants to know the answer to the question has not yet been devised in any mainstream way, but it's interesting to be on the lookout for anything, since I believe personally that there must be some way in which it is not impossible, even if that involves newer forms of experience that, say, technological innovation could bring.

• A few other examples of relatively novel ideas that don't have to be considered extraordinary cases in need of extremely extraordinary evidence:

• There may be no such thing as the concept of "one"/singular points

• There may be no such thing as a discrete location in time or space

• Consciousness may be nonlocal/entangled

• Further reading (in which, funnily, people use evidence itself to try to figure out whether evidence is something we need a lot of in extraordinary cases):
  https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3114207/

• Probing the limits of the quantum world:
  https://www.quantamagazine.org/how-big-can-the-quantum-world-be-physicists-probe-the-limits-20210818/