In my view scientists make experiments and get results. Then, usually they interpret these results (theories). We may agree that experiments' results are objective, at least if they are reproducible. So what is the reason to consider that science is not objective?
Empirically minded philosophers assume that the evidential value of an observation or observational process depends on how sensitive it is to whatever it is used to study. But this in turn depends on the adequacy of any theoretical claims its sensitivity may depend on.
Thomas Kuhn, Norwood Hanson, Paul Feyerabend and others cast suspicion on the objectivity of observational evidence in another way by arguing that one can’t use empirical evidence to test a theory without committing oneself to that very theory.
This phenomenon is called Theory-ladenness of observations :
in the philosophy of science, observations are said to be "theory‐laden" when they are affected by the theoretical presuppositions held by the investigator.
Is this a treat to Scientific Objectivity ? :
Scientific objectivity is a characteristic of scientific claims, methods and results. It expresses the idea that the claims, methods and results of science are not, or should not be influenced by particular perspectives, value commitments, community bias or personal interests, to name a few relevant factors. Objectivity is often considered as an ideal for scientific inquiry, as a good reason for valuing scientific knowledge, and as the basis of the authority of science in society.
According to a popular picture, science progresses toward truth by adding true and eliminating false beliefs from our best scientific theories. By making these theories more and more verisimilar, that is, truthlike, scientific knowledge grows over time (e.g., Popper). If this picture is correct, then over time scientific knowledge will become more objective, that is, more faithful to facts. However, scientific theories often change, and sometimes several theories compete for the place of the best scientific account of the world.
Kuhn's analysis is built on the assumption that scientists always view research problems through the lens of a paradigm, defined by set of relevant problems, axioms, methodological presuppositions, techniques, and so forth. [...] Can observations undermine such a paradigm, and speak for a different one? Here, Kuhn famously stresses that observations are “theory-laden” (cf. also Hanson): they depend on a body of theoretical assumptions through which they are perceived and conceptualized.
In conclusion : experiments' results are objective, when they are reproducible. Yes.
But scientific knowledge is made also of theoretical interpretation of experimental facts and evidence.
Thinking about who decides which experiments are conducted, to whose benefit those results might be, can be a starting point. The peer review process is not perfectly objective. Old scientists refuse to consider new theories (Max Planck said that "science advances one funeral at a time"). Plus you can look at scientific history, especially about racial science/eugenics which was affected by the racist Imperial European politics of the time, and so on. In my undergraduate degree, they told us that science operates in terms of paradigm shifts, where scientists believe in/uphold a certain paradigm and then a combination of political and scientific inputs cause, after a period of stability, a shift to a new paradigm. Anyway, all this is by way of saying that the scientific method may be (or strive to be) objective, but science as practiced is done by humans, and therefore is infected by politics. Or alternatively, it is done situated in a political environment.
Taking a definition for science that presupposes objectivity, then of course science is objective. But that's trivial.
Taking a definition for science that allows human elements, then of course science is not objective: the humans of science are subjective, and so science is subjective.
I believe that the objective definition for science will not be satisfying, because the field will be limited to what's experimentally verified, ruling out most theory.
I think that the subjective definition for science will not be satisfying, because it lacks the authority that comes of absolute truth.
Maybe it would be better to worry less about what science is, and just focus on individual assertions by individual people:
"I did this in the laboratory, and a lot of smart people agree that this is always just what happens."
"Whenever we look over here, we see the same kind of thing. Maybe that's what's causing all this other stuff to happen."
"Most of the time, this experiment goes to the right. But it goes to the left sometimes. Let's build an experiment that tries to determine just when we can expect is to go left."