If not, why doesn't some other army overthrow the current rulers of the Iron Bank, and keep the gold for themselves?
The headquarters of the Iron Bank is located in Braavos, a city-state built on a group of small islands in a shallow sea. As such it is very difficult to attack:
The geography and history of Braavos are clearly modelled on Venice, which was an independent republic for more than 1000 years until it was taken by Napoleon in 1797. For most of that time, Venice was a wealthy trading city with much larger and highly aggressive neighbours, but its shrewd diplomacy, natural defences and powerful navy meant that it was never conquered. We can assume much the same is true for Braavos.
Finally, it is likely that most of the wealth of the Iron Bank is not kept as gold in a vault. Like other banks throughout history, it would have invested its money in land, trading fleets, workshops and other enterprises, which are not nearly so simple to steal.
The Iron Bank itself does not have a large standing army, though their currency repository is pretty well guarded. What's a much bigger deterrent is the fact that the Bank is considered such a stable and universal form of currency that they can essentially "buy" whatever military support they need from anywhere in the world.
The Iron Bank was formed in Braavos before the Doom, when Braavos was a "secret" city, so it was important that everything about the bank had to be kept hidden. They get their name, in fact, from the abandoned iron mine where they lock up the bulk of their currency stores. Since it's buried underground with only a single entrance, it's very easy for them to fortify and guard this repository. As they've grown bigger, they've also been able to use the money from their investments to purchase larger and more secure facilities. That means, anyone trying to overthrow them would have a pretty big hill to climb even if they could do so unimpeded.
More importantly, though, the Iron Bank is the source of financial stability across the entire planet. Again, this largely derives from it's original status as an independent country, not officially part of the Valaryian Freehold and, later, as one of the Free Cities. The Braavosi in general have a strong mercenary streak in their culture, so there is a general feeling that the Iron Bank "stays out of" political conflicts. This makes the bank very important to the various governments across Essos and Westeros. There are other banks in Essos (Pentos and Myr both have one) but those are largely considered "lesser" banks; The Iron Bank is the one that is universally respected and feared by everyone.
That means, among other things, that the Iron Bank is owed of money by nearly every major government on the planet. In a pinch, if the Iron Bank felt threatened, they would simply pay one of the major mercenary forces, or offer to cancel some debt from a country with a standing army, until they had a sufficient force to repel any invaders and/or re-conquer their homeland. An offer of payment from the Iron Bank is about as solid an investment as there is in that world, so they would have had no problem (for example) buying the entire army of Unsullied on credit, or paying the entire Golden Company.