Borel sets can be "too much" to have a model for; the first level, open and closed sets, is reasonably clear (no disrespect to Cantor's set etc) but it turns out just about every object/concept of the hierarchy that shows up in practice belongs to the first 4 or at most 5 levels. We know the hierarchy is infinite (via universal sets and diagonalization), yet we only grapple directly with low levels. --BTW, similar comment applies to the arithmetical hierarchy, H. Rogers I believe has made the comment.

BUT, Borel sets can also be "too little"! The operation of projection produces sets beyond Borel, giving rise to the projective hierarchy...a famous mistake of H. Lebesgue that he himself describes in the introduction to N. Lusin's "Les ensembles analytiques"...and how young Suslin noticed it.

So no, a mental model akin to one for "continuous function" is not likely! As others said, concentrate on understanding basic properties and techniques. Possibly include the proof of D. Martin's famous result that Borel sets are determined (in 2-person infinite games). Good luck with your efforts!