The apparent impossibility of life emerging out of “nothing” is an illusion

“Every single event that can be thought of, has assigned to it a probability which determines what the chances of that event occurring are. All events therefore can happen, but for some events the probability of occurring is infinitely small. Given an infinite amount of time however, what can happen, will happen.”
-Murphy’s law (paraphrased)

Imagine an earth without life on it, no animals, no plants, not even microorganisms, just land surrounded by a giant ocean, dust and inorganic chemicals swimming in the seas. This not so pretty planet revolves around the sun, and gets its daily dose of sunlight, which energizes the chemicals causing them to react with one another; a process which continues for about a billion years until a remarkable reaction occurs! The accidental production of the first replicator; a molecule that carries the innate tendency to accidentally produce copies of itself!

DNA is a replicator, however the first self-copying molecule – the “original” DNA – must have been different, and much simpler than the double helix we are all familiar with. To get a sense of what the first replicator molecule could have looked like, think of a small molecule called A and another one called B. Let’s presume for the sake of the argument that A and B were some of the byproducts of the chemical reactions induced by the sun’s radiation. Molecule  had the tendency to form strong bonds with other A molecules and molecule B would form strong bonds with other B molecules, forming chains of similar blocks – or polymers. The two molecules had also had the propensity to attract each other weakly – at least in comparison to the attractive forces they would generate on molecules similar to themselves (i.e. molecules with which they formed chains with).

Having these two characteristics, when a lot of As and a lot of Bs happened to be found close to each other they would form configurations similar to this:
A-A-A-A
B-B-B-B
For the attraction forces between the two chains is weak, such configurations would get disrupted when subjected to the forces of nature. The two separated chains would then be free to attract other B or A blocks respectively, and aggregate them close to each other, enabling them to react and form chains themselves; introducing with each separation a new copy of the original molecule.
A-A-A-A
B-B-B-B
would break into A-A-A-A & B-B-B-B which would then result in:
A-A-A-A         B-B-B-B
B-B-B-B and A-A-A-A

Now let’s introduce some errors to the process, because nothing is always 100% perfect. Imagine that for some reason you were told to copy an original string of words such as a paragraph 1000 times, by hand, in a notepad. Most probably some of your copies will contain spelling mistakes, in others you might have changed the order of some words a bit, or you might have omitted a few words, but ultimately you managed to keep the context of the paragraph unchanged. Imagine that those copies you created are now passed on to others, who are too asked to make copies of your copies; the copies they made are then given to others and so on. At some point, as you can imagine, the context of the original paragraph is bound to become distorted, as the words become rearranged and omitted, spelling mistakes are propagated and amplified etc. Given enough time, the end result is a huge amount of paragraphs, each different from each other and all having a single common ancestor (the original paragraph). Some of these paragraphs that make no grammatical sense are considered as unstable and are discarded from the “paragraph pool” whereas the rest are more stable and coexist for the time being.
Side note: Food for thought: Ever wondered how it came to be that there are so many religions and sects in this world?

Exactly the same principle applies to our replicators! For instance, something like this could have happened: During the process of chain construction, a third type of molecule called C found itself close to the chain or As, and it turned out that C had similar (but not exactly the same) properties to A, and managed – under the influence of UV radiation – to find itself attached to the chain of As. The C now attracted Bs but also attracted Ds, leading to a new configuration of chains…you can see where this is going. The main point is that the primordial ocean was now populated with multiple types of replicators, which as you can imagine, have access to and compete for the same building blocks (by compete I don’t mean consciously; after all, these are simply molecules who unintentionally make copies of themselves using the blocks of chemicals found around them).

Competition must have caused many replicators to become extinct, and only the most competent ones – the ones most chemically prone to adaptation under the subject of nature’s environmental pressures – get to propagate themselves while time passes (this is by the way the basis of the theory of evolution). At some point, some replicators must have acquired features such as the capability to abstract chemical building blocks from other replicators, or, the ability to protect themselves from such attacks by using some sort of encapsulation, or even the ability to move around using the energy released from chemical reactions, and perhaps the capacity to sense large concentrations of building blocks etc…and thus the beautiful journey of life began, and here we are ~3.8 billion years later.

A couple of things worth noting:
a) The first step to the beginning of life was the emergence of a replicator molecule, an immensely improbable event which occurred only because of the immense amount of time it had to its exposure to occur. If this is hard to comprehend think of it this way: Your chances of winning the lottery are 1 in 14 million. If you could play every day for 1 billion years (approximately the time it took for the first replicator to emerge), you could possibly win ~71 times.
b) Note that the process of the development of life is not a conscious process! It is a process driven by nature and this is what makes it so perfect; because nature never gave second chances to the ones who could not adapt to its pressures and never will. The resulting species that coexist in the present are the best ~3.8 billion years of dynamic adaptation have to offer. Life as we experience it now is not so improbable after all!