This is a bit late, but I disagree with Duncan McKenzie's answer. Great platonic relationships come out of wanting to stay friends and not start anything romantic. Obstacles aren't everything, either; unless it's supposed to be a love story (and even sometimes if it is), one good external or internal obstacle is the most you'll ever need.
On the note of how to write a platonic relationship, there are a ton of really good examples out there. My favourite of all that I've seen would have to be Roy Mustang and Maes Hughes in Fullmetal Alchemist. They definitely share deep love for each other, but it's not romantic at all. They care for each other like brothers and lovers and best friends and so much more that can't be categorized or neatly put in a box. They each understand what the other is thinking and feeling, and they watch out for one another. They bring out the best in each other, and they're the only ones who can really bring out the worst (the angriest they ever get is when someone hurts the other).
Now, maybe you're not looking for something quite that deep. That's fine. In that case, I'd follow Lauren Ipsum's advice: Look at your own friendships and build off of them. That way will almost invariably give you a fantastically realistic platonic relationship that your readers can usually relate to with ease.
This part applies to any level of friendly relationship, and it's mostly just my personal preference. When reading and writing stories with characters who have known each other for some time, I find that some of the best material comes from the banter between them. You know, the little inside jokes, the running gags, the spurious insults, and stuff like that. For example, I recently read a story where the two main characters had a running game where they would scream and rage at each other over the smallest things ("How DARE you borrow my pen, you soulless son of a bitch!") or dramatically declare their undying love for one another in "raunchy, hyperbolic detail", with the point of the game being that they had to remain completely serious no matter how far it escalated or how ridiculous it got. This little game turned two otherwise serious characters into children for a paragraph or two and added some much-needed comedy to an otherwise dark and horrific story. Point is, it's the little things like that that really give a relationship dimension and intimacy. Think of the things that you and your friends do that annoy others and/or that others wouldn't understand.
Additionally, having these little things develop over the course of a story is a great way to show the growth of a relationship between some characters.
Honestly, having them interact a lot in general in a realistic and intimate fashion (no, not necessarily in that sense, just in the sense of making sure that the reader can tell that they have a deep connection) can go a long way. Showing that they can usually tell what the other is thinking or feeling is a great way to show that they're "more than friends" in the literal sense of the phrase.
The beauty of a strong platonic relationship that becomes romantic is that you don't even have to lose the platonic aspect of it at all. Sure, things are different between them, but they're not any less friends because of it. In fact, it can even deepen their bond, just adding another layer to what makes their relationship so special. The banter continues, their friendship still shines, and the romantic aspect of their love never completely takes over (unless you're turning the book into a romance, of course). Think of Chandler and Monica from Friends, or Elizabeth and Mr. Darcy from Pride and Prejudice. Even after admitting their feelings for one another, they don't just suddenly turn all lovey-dovey. They keep cracking jokes and teasing one another, just as couples as well as close friends.
Hope this helped and wasn't just a wall of text.