Have philosophers explored the ineffable, and what in our language makes it impossible to describe something beyond our realm of existence?
There might be things outside of the universe we can't access and that can't be described, because our senses cannot perceive or our minds cannot even grasp the things outside of it. Have metaphysical philosophers explored the possible reasons why our language wouldn't be able to describe these?
The Perennial philosophy would be what you're looking for. Many philosophers explore and write about the reason why Reality is not just ineffable but beyond conceptual fabrication. It would be because Reality outruns the categories of thought. For this reason mysticism asks us to explore beyond the intellect.
We are speaking here of the Biblical Tower of Babel and the reason no construction of logic can reach all the way to Heaven, the impossibility of 'idolising' the Ultimate. I cannot suggest a simple way into the topic. It is discussed by Nagarjuna. Plotinus, George Spencer Brown, Charles Peirce, Lao Tsu , Nicolas de Cusa and countless others but nobody makes it an easy topic.
On the other hand, it is easy in a way. Nicolas de Cusa explains the difficulty arises because the Ultimate or true nature of Reality lies 'beyond the coincidence of contradictories', which is to say beyond the categories of thought. Clearly we cannot think beyond the categories of thought.
Lao Tsu also briefly summarises the whole problem when he remarks 'True words seem paradoxical', which is the case because of what de Cusa said. But to see the topic in this simple way requires a study of what lies behind these words, and for an intellectual grasp of it Nagarjuna may be your best bet.
If you're seriously pursuing this question then I'd recommend The Sun of Wisdom: Teachings on the the Noble Nagarjuna's Fundamental Wisdom of the Middle Way by Khenpo Tsultrim Gymaptso. If you haven't done so already then before reading this I'd suggest examining the reasoning that led Kant to his indescribable 'thing-in-itself'.
EDIT: In respect of the question about language I just remembered Bradley. He blames the subject-predicate form of language for its inability to reach into the depths of metaphysics since it divides the world. His Appearance and Reality is a good read.
There are two slightly different flavors to 'ineffable', let's call them 1) things impenetrable to our understanding and 2) things that defy description. For the latter we need not venture beyond our imaginations to find the limitations of language...
Fortunately there are ways to supplement natural language, let me try to explain with this handy metaphor :) by MC Escher:
We are tempted to thinks of words as mapping, one to one, with objects of experience: this is just the way we use it on daily basis. However, given there is much difference of opinion on exactly how, language is more like one hand in the drawing. It describes reality (note the atomic word 'scribe'); and it does so in a construtivist and at least semi-consensualist manner. Language grows, but at the same time, together with our understanding, Reality (seems to) grow. Thus the descriptor always lags in denotative power.
Now let us suppose that the two pencils aren't exactly the same (the reality of Language is that it isn't perfect), maybe slightly different color. We can see the description of Reality is close enough to be isomorphic (it is the same picture from different perspectives), except one aspect, the color, is not only different but intrinsically incapable to be produced by the descriptor (Reality must change for Language to gain that capacity).
Yet another aspect not so easily put in words but appreciable by the fact that there are two hands "necessary" to convey the concept: Let me echo a quote from @PeterJ answer.
Reality lies 'beyond the coincidence of contradictories'
To wrap our minds around something we often employ dichotomies. I.e. we talk about 'warm and cold' when in reality we a referring to a physical quantity temperature. To conceptualize heat we first relativized it then quantized it. Language builds upon understanding which builds upon language... But 'warm' and 'cold' is not from nature, their axis is the coincidence and they're the contradiction.
As for things we simply can't understand: obviously for some we merely need the language to develop. Obviously for others there is a "third perspective" forever out of reach.
See also:
As fantasy fiction illustrates, it is very possible to use language to describe "something beyond our realm of existence".
However, this does not mean that everything we experience can be explicitly described using language. As Michael Polanyi notes, "...all knowledge is either tacit or rooted in tacit knowledge." (page 7) Tacit knowledge, such as how to drive a car, is not able to be made completely explicit.
This suggests that it is possible for language to explore the ineffable in at least some round-about way. What does seem to be impossible is to completely express in language even ordinary, non-ineffable events.
Polanyi, M. (1966). The logic of tacit inference. Philosophy, 41(155), 1-18.