I am a bit confused of the "concept" of irreality in Konjunktiv-2. Suppose at some point I say
Ich denke, ich würde zu einer Bibliothek gehen
and then later I infact go to a library, would this make the usage of Konjunktiv-2 before unjustified?
The point is, in the moment of me thinking it, I don't know if it happen or not. The idea is sort of in a indeterminate state. So, does konjunktiv-2 make sense here?
Ich denke, ich würde zu einer Bibliothek gehen.
This sentence is semantically saying something about the future, but grammatically it is Präsens. These versions are grammatically future tenses:
Ich werde denken, ich würde zu einer Bibliothek gehen.
Ich werde gedacht haben, ich würde zu einer Bibliothek gehen.
Again, semantically, this sentence is saying something about a possibility with a low likelihood. But grammatically this is a "würde-Ersatzform". These versione really uses Konjunktiv II:
Ich denke, ich ginge zu einer Bibliothek.
Ich werde denken, ich ginge zu einer Bibliothek.
Ich werde gedacht haben, ich ginge zu einer Bibliothek.
Now about your question:
Irrealis means, that in the moment when you say it, it seems impossible. This is all what counts. It is irrelevant what really will happen later. The current expression of the future can never influence an event in the present. If that were so, the cause would be temporally after the effect. This is not only physically impossible, but also linguistically.
Instead of trying to master the concept first, I would commence with the contexts in which Konjunktiv 2 is used. Among these are:
Konjunktiv 2 in these examples seems to be stronger than what the term irreality might suggest (if it was understood as unreal or insubstantial): it affirms the truth of the negated proposition, e.g. wenn du mir doch nur zuhören würdest presupposes that du hörst mir nicht zu is true.
When talking about the future, German uses either the present indicative or werden plus infinitive.
Morgen sehe ich mir einen Film an.
Morgen werde ich mir einen Film ansehen.
Such sentences will usually be interpreted as talking about an intention or expectation, with all the caveats that apply to that (intentions may change, expectations can be disappointed), which is in a way weaker than what Konjunktiv 2 expresses with is "counterfactuality".
With Konjunktiv 2, there is no contrast between present and future.
Was mir gerade helfen würde, wäre eine Umarmung.
Was mir morgen helfen würde, wäre ein Umarmung.
With the indicative, there is a choice.
Was mir morgen hilft (or) helfen wird, ist eine Umarmung.
Konjunktiv II does not express irreality in the sense that the statement is not true or never will become true. For that, negation would be used: Ich gehe nicht in die Bibliothek.
Use of Konjunktiv II is restricted to a number of cases. David Vogt listed some of them in his answer.
When used in a main clause, Konjunktiv II is used to express that the statement would be true under a condition which is not satisfied.
Without such a condition, use of Konjunktiv II in a main clause (or, in the example at hand, an Objektsatz) does not make sense. So if I read the sentence Ich denke, ich würde zu einer Bibliothek gehen I would expect that there is some (not explicitly mentioned) condition under which the speaker would go to a library but which is not met or which is at least uncertain. Without such a condition, the sentence would hardly make any sense.
However, it could be that the condition is met later. In this case, the sentence does not become invalid when the speaker goes to a library later.
Your example starts a bit late for an easy explanation. It seems to be the answer to a question like:
Wenn du ein Referat über xxx schreiben müsstest, wie würdest du vorgehen?
(Engl.: If you had to write an essay concerning xxx, how would you proceed?).
This is a hypothetical situation, which can't easily become reality by your stroll to the library. (Apart from the fact, that the search engine of your choice is the much more likely answer today.)
