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In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or physical grounds.

Are there applications where non-separable Hilbert spaces naturally arise?

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The main example of a non-separable Hilbert space is the Besicovitch space of almost periodic functions. Almost periodic functions play a significant role in analysis, from differential equations to operator algebras, and this space is quite useful.

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Here's Besicovich book (I was giving the same answer...) plouffe.fr/simon/math/… –  Pietro Majer Aug 8 '13 at 7:19
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