The SEP article on Donald Davidson gives an overview of his outlook on this question:
Davidson’s commitment to the rationality of the mental as one of the cornerstones of anomalous monism (as well as to the account of ‘radical interpretation’ [see ‘Meaning and Truth’ below]) led him to take a special interest in the problem of apparently irrational belief and action – something first addressed in ‘How is Weakness of the Will Possible?’(1970a). While Davidson treats irrationality as a real feature of our mental lives, he offers a way of dealing with it that aims at preserving, in some sense, the overall rationality of the mind (see especially ‘Two Paradoxes of Irrationality’ [1982b]). A belief or desire in the mind of one person can cause a belief or desire in the mind of another without this compromising the rationality of the mental. (Davidson’s example is my growing of a beautiful flower because I desire you to enter my garden – you develop a craving to see the flower as a result of my desire and my desire has thereby caused, without being a reason for, your craving). Davidson suggests that we should view the same sort of relation as sometimes holding within a single mind. To this end we should view the mind as weakly ‘partitioned’ so that different attitudes may be located within different ‘territories’ and need not, therefore, be taken to come into direct conflict.
I mention Davidson because he advanced a seemingly helpful distinction between all-things-considered and all-out judgments:
The phrase “all things considered” is not, as it might seem, merely a minor difference in wording that allows weakness of will to get off on a technicality. Rather, that phrase marks an important contrast in logical form to which we would need to attend in any case in order properly to understand the structure of practical reasoning. For that phrase indicates a judgment that is conditional or relational rather than all-out or unconditional in form; and that difference is crucial.
Now, so far as the OP question goes, we would be assimilating holding-an-irrational-belief to an act of mental will that can exhibit such weakness. I.e. if it is possible to stop at an all-things-considered belief even when a more categorical thought is in sight, then we can at least willfully hold suboptimally rational beliefs.
Another good SEP article to look over in this connection is one added just the other day, on structural irrationality. The idea is, roughly, that we can be irrational either procedurally (how we evaluate evidence/information) or per the content of reasonability (we might be e.g. consistent with our applications of eldritch astrology but appealing to such an antiquated doctrine is prime facie deficiently reasonable). So we can refine the OP question as, "Is it possible to be both structurally and substantively irrational with respect to some belief?"
What we will end up having to look into more, then, is the distinction between "merely non-rational" and "actively counter-rational" beliefs. Moreover, we will have to address our beliefs-about-those-beliefs: if we "know" that some belief is underdetermined by the required evidence, or if we don't "know" that some belief goes against that evidence, then is it possible for us to hold to such beliefs? Or does recognition of suboptimality or outright contrariety, here, indeed block us from continuing to hold such beliefs? This is, ultimately, Immanuel Kant's optimism about the problem: he says that a radically corrupted will, a will going directly and strictly against reason, is not possible for us. So though he laments that we will never be quite free of transcendental illusions, and hence there will always be a temptation for us to end up with irrational beliefs (even contradictory, or antinomian, judgments as such), yet so long as our apprehension of the illusion is clear, we will perforce fend off falling prey to full acceptance of these illusions.
