Relating to my previous meme post, I thought I'd rather explain myself than just leave it at that. I would like to add that these conclusions can be read both platonically AND romantically, and it only depends on how you read the characters - the connection they both share, however and either way, is immense.
"Holmes and Watson practically saved each other's lives". What did I mean by that? Let's take a look at the characters prior to/during A Study in Scarlet to answer that question.
Sherlock Holmes is a 20 something year old who created his own job. He is very competent at what he does, but he never receives the respect of anybody else. Even worse than that, he's probably used to feeling like people around him don't really like his presence among them, as can be perceived once you realize his readiness to call himself out on all of his flaws once the doctor showa interest in living with him. He overworks himself, and probably gets as sick as he gets when the doctor comes around - only this time, he doesn't have anyone other than himself to rely on. He already does substance abuse to cope with his fits of depression (cocaine) and, most likely, mania (morphine). Following this life the way that he was, he would most likely end up dead at a very young age and would never get the recognition he so thoroughly believed himself capable of. He may be a cheerful man, but it still doesn't break the fact that it is quite tragic to see the "genious detective" passing by like a whispering wind too quiet to have it's presence perceived.
And then dr. Watson appears, and he admires him enough to stick around with him even at his worse. Holmes tries behaving himself when they meet, he writes an article and circles it just so the doctor will ask him questions of his expertise, he fears he's going to be uninspiring if he overly explains himself... and then Watson replies by stating he is very impressed by how he made his own science. All of this happens in A Study in Scarlet, and you can see how much Holmes appreciates his flatmate's praise when he gets annoyed by Watson's "dismissiveness" in The Dancing Men ("How absurdly simple!", he says, but he doesn't mean to sound dismissive; Holmes only perceives it as so and it is enough to make him upset for at least a short while). Dr. Watson also takes care of the other part of Holmes' that could potentially turn his story to a more tragic area: his health. He isn't a reckless man overall, but in the subject of physical health he can be rather unconsequential. You learn from The Norwood Builder that fainting due to inanition is common when Holmes is working, you see Holmes illness being a plotpoint in at LEAST 2 stories, he doesn't protect himself on the laboratory in A Study in Scarlet and ends up with scarred hands, and you see everytime how easy it is for him to neglect himself for the sake of another person's life (a client). Dr. Watson, quite literally, saved his life - and gave him the fame he knew himself to be capable of getting, although he never used the highlights for his own ego - he reached more people this way, and therefore did more things and felt more useful.
On dr. Watson's end of the stick: he just arrived from war. He is permanently injured in at least two spots, and you can see his disability makes him feel worth less than before in The Sign of the Four. He saw Death with his very own eyes, and surviving doesn't seem much better; to his eyes, he's a broken man in every sense of the word, including the financial. He says he's been "living a meaningless sort of existence" in the radio adaptation, and I believe this to match how he viewed his life post-war for some time. There is no way to know what would have happened to him, since he doesn't strike the reader as a neglectful man in the same sense Holmes does, but there's no way to deny that he is depressed over all that has happened and it's results and that there are one or two outcomes to his depression that are as drastic as Holmes dying a nobody.
When he met Holmes, he had nothing to even life for. He himself admits he had so much freetime on his hands, he turned the observation of his flatmate into a personal hobby. And that's how Holmes gave his life emotion, through all of his cases and mysteries. This extraordinary man cared about him, Watson noticed, and coloured what was long gray. He was one of his detective heros, Dupin and Lecoq, only in flesh and bone and improved - and he cared about him. Can you imagine what it feels like to meet a person that matches a fictional thing that has brought you comfort many times before, and this person actually turns out to like you?
Upon seeing Holmes' anonomity, Watson was the person who gave him fame. He wanted to do him a favour when he did that, because he thought it was unfair he spent so long studying and perfecting himself only to be ofuscated by other investigators. Holmes may not like the way that he wrote the story, but you can see his acknowledges it enough to mention his writings here and there, so he at least reads them - but I'm going off topic.
The point is: they saved their lives both metaphorically and literally, because each of them gave the other a sparkle that was missing before. And wether you see this as platonic or not, it is still very nice to see how they genuinely enjoy each other's company.
