It may seem hard to add a new answer to all this, but here's mine. How to motivate the open set garbage of topological spaces:

Answer: Don't.

There are many ideas in mathematics that can be easily derived from some real situation, and I would count approximation (ie limits), metric spaces, and neighbourhoods as among these. I think that it is quite easy to motivate the neighbourhood definition of topological spaces, for example, by considering real world examples of needing approximations that can't be controlled by metrics (for example, if you always need your approximations to be greater than the true value).

But one can take this line too far and try to motivate everything in mathematics from real-world situations and this, I think, misses a great opportunity to teach something that all students of mathematics need to learn: that when something is presented to you in a particular way, you don't have to accept that viewpoint but can choose a different one more suited to what you want to do.

We try to teach them this with bases of vector spaces: don't use the basis given, use one that makes the matrix look nice (diagonal if possible!).

So here, we can present topological spaces as sets with lots of declared neighbouhoods satisfying certain simple, intuitive rules. But they are hard to work with so instead we work with open sets (sets which are neighbourhoods of all their points) because it makes life easier.

I should qualify the above a little. It's written as a counterpoint to all the previous replies which try to justify open sets based on some intuition. I'm not saying that those are wrong - far from it - just that with something like this, one should think carefully about the message one is sending to the students about mathematics. At some point, they have to learn that mathematics strives to be clear and elegant rather than intuitive and vague, and it's a good idea to do this with an example like topological spaces where we are still close to the intuition, rather than something like function spaces where intuition often takes a hike.