I’ll take a stab at this, though as will become clear as you read my response I don’t think this is the sort of question that admits of a simple answer.
So, in order to decide if history is a science, we would seem to need to know what a science is. Or, at the very least, we need some set of necessary and sufficient conditions for when something gets to count as a science. This sort of thing is what is known in philosophy as a demarcation problem. The demarcation problem in science has gotten a lot of attention, which I will briefly summarize here.
First, you might think that science is simply those studies that deal with what is objectively verifiable. To be a bit more careful, we might say something like this:
X is a science if and only if X deals with what is empirically verifiable or analytically certain.
This sort of view is called verificationism and is most closely associated with the Logical Positivists of the early 20^th century.^1 Let’s unpack this a little. First, “analytically certain” is meant to allow for science to deal with what we might loosely term definitional truths. Stuff like “All bachelors are unmarried men” or “1=1=2” and so on.
What about “empirically verifiable”? Well, clearly we don’t want to require that science be empirically verified, since we are doing science before we get the results of experiment. We also don’t want to say that science must eventually be verified, since Newton’s theory of gravitation is clearly science but could never be totally verified given that the theory turned out to be false. So what we seem to want to say here is that science deals with theories that could, in principle, be verified empirically.
Ok, so what is the problem with verificationism? Well, consider the case of astrology. Take the claim: Capricorns tend to be angry. Now, this claim is obviously, in principle at least, empirically verifiable. We could go take a poll and find that Capricorns do, in fact, tend to be angry. But surely we don’t on such grounds want to say that astrology is a science! So verificationism seems to get these sorts of cases wrong.
Another theory, due to Karl Popper, is that science deals with claims that we could in principle falsify. Such a view, naturally enough, is called falsificationism. We can deal with this theory quickly, since it falls prey to the same counterexample as verificationism. Again, clearly we can empirically falsify astrological claims, but this does not mean we want to say astrology is a science.^2
The best view, at least imo, is that science is what Wittgenstein called a game concept.^3 Roughly, science is an organic concept which, while meaningful in ordinary language, does not admit of necessary and sufficient conditions. Rather, it is defined by usage and paradigmatic instances. So, we might say that science is defined by how we use it to classify paradigmatic cases like physics (it is a science) and astrology (it isn’t).
Unfortunately, history seems to be a borderline case. It doesn’t appear obviously to be a science like physics, but it also clearly doesn’t belong with non-sciences like poetry or astrology. At the end of the day though, I don’t think this is so bad. While we tend to attribute a ton of value and authority to science, we needn’t do so to the exclusion of other fields. And fwiw, history clearly seems closer to a science than not, imo. History is informed by the empirical investigation of the world (via archaeology, historical records, etc) as well as careful appeals to authority (e.g. when a Roman historian appeals to Tacitus). History emphasizes rationality and checking oneself carefully against the evidence. Maybe it is a science and maybe it isn’t, but it certainly seems to me to be both rational and valuable. That is good enough in my book.
Notes
This school is perhaps typified by figures like AJ Ayer. Awesomely, you can actually hear Ayer himself talk about Logical Positivism in this video.
Obviously, this is way, way too quick. For a good overview of Popper’s view of the demarcation problem, see here and sources cited therein.
Again, since this is a reddit post, I’m moving through this rather too quickly. See here for more details.
The real question is ask is, in what context are we caring about the answer to the question? What are the stakes?
If the question is, "which department of the university should they be classed under?" historians look, from a university perspective, much more like humanities than sciences. We don't (usually) needs labs, for example, and our funding requirements are more like other humanists (go to conferences, fly to a place to do research, time to write books) than scientists.
If the question is, "is the knowledge more reliable than what we think of as humanities?" then that gets you into a whole other bucket of worms. Historians are empirical, unlike a lot of the humanities (much of which is aesthetic), but our methodologies and goals are far looser than those of the sciences. Our primary output is descriptive narrative, like much of the humanities, even though it is non-fiction. We explicitly do not pretend our truths are universal in the sense that scientists think their truths are universal — we embrace subjectivity and interpretation much more than they do.
For what it is worth, I think history straddles boundaries that exist between the traditional humanities and the social sciences. It can depend on the flavor of history (there are many different types), it can depend on the context. For what it is worth, my own experience, as a historian of science at an engineering school, is that I find the true "two cultures" problem not to be between engineers and humanists, but between empiricists (people who use evidence to come to conclusions) and aestheticists (people who are primarily interested in a subjective experience). In this dichotomy, historians are firmly empiricists — we can talk to social scientists, scientists, engineers, etc., and everyone can fully understand one another's goals, methods, etc., without any serious amount of translation. Whereas talking with "true" artists is an entirely separate thing — they have almost incommensurable goals and methods. But this might just be my experience (and the particular artists who I work with — and I should say, I do actually work/collaborate with artists at times, even if I'm not sure we all fully understand one another!), and my own inclinations.
Historians are methodologically unruly; we have a few basic things we agree on, but every historian does things a bit differently, in a highly-individualized mixture of the empirical and the subjective. That's a feature, not a bug, in my mind — it allows us to produce investigations of the past which transcend mere chronology, that get at something more profound, interesting, and ultimately useful. But in the end, we read dead people's mail for a living and use it to write stories; it's a fundamentally strange enterprise. But the world is a strange place.