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.