Kierkegaard, despite many claims to the contrary, was not an irrationalist. He did not think that one can believe whatever one wants to believe, and his famous claim that ‘subjectivity is truth’ was never meant to convey that truth is simply relative. In fact, Kierkegaard clearly indicates in both Philosophical Crumbs and Concluding Unscientific Postscript that objective knowledge (in certain areas) is entirely attainable. In particular, he references the spheres of mathematics and logic, where absolute (ideal) knowledge can be attained. He also affirms at least approximate knowledge in the case of empirical observation and, to some extent, historical knowledge where evidence is available.
When Kierkegaard claims that subjectivity is truth, his main goal seems to be to claim that objective knowledge in the ethico-religious sphere is impossible. But this is not necessarily because ethical or religious truths are not objectively (in themselves, without reference to anything else) true. Rather, those kinds of truth can only be known subjectively. Human beings cannot know those truths objectively because knowing those truths objectively requires a disengaged, abstracted stance. The problem with that stance is that it is impossible for humans to hold such a stance while also living in the truth of the ethico-religious claim. Let’s take an example from SK (through his pseudonym Johannes Climacus) directly:
If one who lives in the midst of Christendom goes up to the house of God, the house of the true God, with the true conception of God in his knowledge and prays, but prays in a false spirit; and one who lives in an idolatrous community prays with the entire passion of the infinite, although his eyes rest on the image of an idol; where is there most truth? The one prays in truth to God, though he worships an idol; the other prays falsely to the true God, and hence worships in fact an idol. (Concluding Unscientific Postscript, 179-180)
This understandably makes contemporary Christians squeamish, because most of us have been taught that our faith is built around having a right, true, objective conception of God. We must objectively know that the God we worship is the true God, otherwise our worship is meaningless. This stance fundamentally misses at least a few points. First, can our conception of God ever be accurate? We may find some way of approximating something about God, but in the long history of human attempts at understanding God, the best we can come up with is that God is infinite (a relatively meaningless concept in everyday life), God is love, etc. If we were required to objectively know the God we are attempting to worship for our worship to be valid, wouldn’t all of our worship necessarily be invalid?
Second, Kierkegaard is not denying the objective existence of God (or anything else for that matter). He specifically mentions in the passage above that the one living in Christendom has a “true conception of God.” His project, at least through the Climacus writings, were an attempt at making hard distinctions within the spheres of human knowledge. There are things that are knowable by humans objectively, but there is also a hard limit on what can be knowable that way. When we attempt to know ethical or religious truth objectively (i.e., cross the knowledge boundary), we make a category error. Further, humans do not have the ability of holding on to knowledge objectively without being dispassionate and disengaged, and this is exactly the wrong stance within ethical and religious truth. Therefore, Kierkegaard doesn’t deny objective truth directly, but the function of holding on to knowledge objectively. Function and existential application were his primary concern.