The term "absurd hero" was coined by French philosopher Albert Camus in 1942 in his seminal work "The Myth of Sisyphus". The book contained a series of essays exploring his philosophies on the nature of existence as part of a movement later named "Existentialism". (Side Note: Camus was techinically an absurdist as that was the movement he invented in an attempt to separate himself from the existentialist movement. Despite this, most people consider Absurdism an off-shoot of Existentialism rather than its own separate philosophy).

The myth in itself tells of Sisyphus's plight as he is sentenced to eternally roll a rock up a hill, watch it fall, then start it over again. He is doomed to do this forever. As Camus says, Sisyphus is an absurd hero "as much through his passions as through his torture. His scorn of the gods, his hatred of death, and his passion for life won him that unspeakable penalty in which the whole being is exerted toward accomplishing nothing." By now the parellel between Sisyphus and Shirou may have started taking shape. Shirou dedicated himself to an impossible ideal, endlessly being used by humanity to maintain the balance and deal with on-coming trouble. Archer, one version of Shirou who reached this point, crumbled under the weight of it all and gave into the absurdity. In the eyes of Archer, his entire existence amounted to nothing. His life (and perhaps all life) was utterly meaningless. In a desperate attempt to rectify his mistakes he is summoned to his own era, many years before he became the fabled "hero of justice", to try and kill his own self. In other words, to commit suicide.

Anybody who has heard of Camus knows the famous opening line to "The Myth of Sisyphus": "There is but one truly serious philosophical problem and that is suicide". If there is no meaning to life, would it not be better to end one's life? That would be the logical conclusion of that trail of thought. However, one thing "Fate/Stay Night" sets out to do is to draw a distinct line between logic and ethics. The term "right" or "correct" is not singular in meaning. In "Fate/Stay Night" there are two forms of "right": A logical form of being right, and an ethical form of being "right". This is shown through the famous quote during the twentieth episode of the TV series, in which Shirou criticises Archer that "just because he is correct, doesn't meant that [he's] right". Whilst to some this may seem like a nonsensical line, it sums up one of the core concepts of "Fate/Stay Night's" philosophy: just because it is the most logical answer, doesn't mean it is the right one.

In fact, one of the core themes of the series, that of fate, is in fact directly linked to this concept. "Fate/Stay Night" offers a constant debate around the idea of logical fatalism, constantly questioning the absoluteness of ethics and life. An example of this would be Kuzuki's character. Specifically, a scene in the sixteenth episode that is exclusive to the blu-rays where Kuzuki and Archer engage in a debate about their reasoning for why they are doing what they are doing. Kuzuki claims that he has zero indication of what it means to be good or evil and down to the fundamental level of his existence is neutral. If somebody has no direct involvement in his life then he doesn't care what happens to them. In fact, Kuzuki claims that it is inevitable for people to kill others; an opinion Archer refers to as "Pessimistic Fatalism" (however, Kuzuki goes on to refute this, claiming that he is neither pessimistic, nor believes that the future is set in stone). Kuzuki's character works as an opposition against Logical Fatalism, which states that something is either true or false. There is no inbetween; there is no grey-area.

So, to what end does "Fate/Stay Night" explore this concept of Logical Fatalism, how does it affect Shirou as a character, and what does Camus have to with it all? It's all to do with Shirou's ideals and past. Ten years prior to the events of "Fate/Stay Night", Emiya Shirou was the sole survivor of a huge fire, saved by the likes Emiya Kiritsugu, his foster father. Throughout the series we never see anything of Shirou's life before the fire and we never learn what his family name was before hand. In a sense, Emiya Shirou and the Shirou before the fire are two separate entities with the fire being Shirou's rebirth in a sense (this isn't explored too much in UBW but in the Heaven's Feel route of the visual novel). Emiya Shirou goes on to live an empty life. He has no true passions of his own and never feels genuine happiness or joy. Instead, he clings on to his foster father's ideal of heroism as his reason for living. It isn't until Archer confronts Shirou that he is forced to examine this ideal and what it means to him. Archer, who suffered at every turn in an attempt to maintain this ideal, eventually gave into logic and deemed his ideal as wrong for the sole reason that it was impossible. It achieved nothing. So his only option was to erase his past self to avoid that fate and end his existence. Despite learning all this, Emiya Shirou refuses that following his impossible ideal is wrong. It may be wrong in a logical sense but not in an ethical one. Logic represents Camus's absurdity and Archer represents the logical man giving into it. Because of this, Shirou (and Camus) could never accept Archer as this "suicide" and regret is "acceptance at its extreme", admitting that life is too much for a person to accept. To accept his ideal as wrong is to admit that Kiritsugu was wrong to save him; that Shirou was better off dead, and that those who died in the fire were right to do so.

Following this logic, the fight between Shirou and Archer therefore represents the birth of Camus's absurdity. The battle is symbolic of the "confrontation between the human need and the unreasonable silence of the world" that Camus claims absurdity stems from. Humans are not beings of logic but of irrationality. Therefore, absurdity "is not in man nor in the world, but in their presence together...it is the only bond uniting them". This, then, is Shirou's answer: whether his ideal is possible or childish is irrelevant - all that matters is that he does what makes him happy and what he believes to be right. Even if it is impossible to be save everyone, that will never mean that wanting to is incorrect. That is cold logic, relying entirely on absolutes, and life is more than that. Shirou's words of revolt to Archer is saying that this "heart within me I can feel, and I judge that it exists. This world I can touch, and I likewise judge that it exists. There ends all my knowledge, and the rest is construction". Does this not parellel Shirou's ability of projection? By visualising the swords in his mind, by using what he sees the weapons as, he applies that knowledge to constructing his blades. I believe if Camus were to describe Shirou in one word, he would call him an "artist". He brings forth what is in his mind and gives it physical form. Shirou "commits himself and becomes himself in his work" as it is his nature that makes him a mere tool for his ideal (the "bone of [his] sword" if you will).

To wrap this up, it may be fitting to glimpse briefly at Shirou's reality marble and his incantation. The most famous translation of the incantation (not the English one Archer uses but the Japanese one Shirou uses) is:
I am the bone of my sword
Steel is my body and fire is my blood
I have created over a thousand blades
Unaware of loss, Nor aware of gain
Withstood pain to create weapons, waiting for one’s arrival
I have no regrets. This is the only path
My whole life was unlimited blade works
Compared to the literal translation found on the typemoon wiki:
His body is made out of swords
His blood is of iron and his heart of glass
He survived through countless battles
Not even once retreating
Not even once being victorious
The bearer lies here alone
Forging iron in a hill of swords
Thus, my life needs no meaning
This body is made out of infinite swords

What sticks out the most is "my life needs no meaning". This perfectly sums up Camus's ideas on life: life is meaningless but that doesn't matter. We do not need a meaning to live but should instead revolt against the world's absurdity in order to feel fulfilled. But what may be the strongest evidence for Camus's influence on the Unlimited Blade Works story would be this passage from "The Myth of Sisyphus":
"I leave Sisyphus at the foot of the mountain! One always finds one's burden again. But Sisyphus teaches the higher fidelity that negates the gods and raises rocks. He too concludes that all is well. This universe henceforth without a master seems to him neither sterile nor futile. Each atom of that stone, each mineral flake of that night-filled mountain, in itself forms a world. The struggle itself towards the heights is enough to fill a man's heart. One must imagine Sisyphus happy."
Is it not fitting, then, that Archer's own "world" is that hill of swords? Each sword, each projection, each battle comes together as the ultimate symbol of Emiya's revolt against absurdity. If Shirou is an "artist" in the eyes of Camus then so is every hero in human history that lived out their ideals and made their mark.

"The present and the succession of presents before an ever conscious mind, this is the ideal of the absurd man". Words from the man himself. An artist who cements himself in the narrative of human history is the perfect absurd hero, living not those few years he is given but eternally as an example of human tenacity and perseverence. It explains just why the Heroic Spirits in the Fate universe exist outside of the shackles of time and live forever across all eras: through their revolution against the absurd, they transcended human limitation and became infinite. However, no one revolted harder than Emiya Shirou, the counter-guardian. To use his own words: his whole life was unlimited blade works.

"All that remains is a fate whose outcome alone is fatal. Outside of that single fatality of death, everything, joy or happiness, is liberty. A world remains of which man is the sole master. What bound him was the illusion of another world. The outcome of his thought , ceasing to be renunciatory, flowers in images. It frolics - in myths, to be sure, but myths with no other depth than that of human suffering and like it inexhaustible. Not the divine fable that amuses and blinds, but the terrestial face, gesture, and drama in which are summed up a difficult wisdom and an ephemeral passion" - Albert Camus, The Myth of Sisyphus (pg. 87)

TL;DR: F/SN is a subtle but well executed musing on the nature of existence and the line between logic and ethics. Emiya Shirou is a representation of Albert Camus's absurd hero and philosophy of life's lack of meaning.