a finite dimensional hilbert space is just R^n, whose points are finite sequences of numbers of length n, equipped with the usual dot product. Notice each vector has finite length, e.g. the squared length of (x,y) is x^2 + y^2.
a typical infinite dimensional hilbert space has as points certain infinite sequences of real numbers (x1,x2,...) but we want a dot product here too, so we set the squared length equal to the infinite sum x1^2 + x2^2 +...
of course here this infinite sequence may not have a finite sum. so we restrict attention to those sequences which do have a finite squared length. this subset of the space of all infinite sequences is a (separable) hilbert space.
so in an infinite dimensional euclidean space, most points have infinite distance from the roign, so we consider only those at finite distance from the origin. that's hilbert space.
there are then generalizations with higher (infinite) dimension as well, whose length is defined by some integral being finite.
e.g. take all functions on the unit interval whose square has finite integral. or all functions on the real line with that proeprty.
now that i reread matt's definit0on, of cousare the abstarct evrsion is that there is adot product,. hence defining a distance, and we require all cauchy sequences in this distance to converge. i hope my examples do this, i believe they do.