The simplest metric, and probably the most frequently used (even if only implicitly), is to count the number of steps between the chords' roots along a one-dimensional line of fifths (or the circle of fifths, if you permit enharmonics and modular arithmetic). I say this is the most frequently used because chord progressions where the root ascends or descends by a fourth or a fifth (which have a distance of one by this metric) are the most frequently used in many styles of Western music, indicating that these chords are "close" in some sense. In your progression, this metric would yield distances of 3, 1, 1, 1. This metric has the property that it plays nicely with tonal music, since the tonic/dominant relation that defines tonality has a distance of one. And even if you start to stray slightly from chords that are in your current key, you will end up visiting "closely-related" keys. Also, because this metric looks only at roots, it doesn't inherently care about chord quality (major vs. minor) or extensions (sevenths, ninths, etc).
Dom has already mentioned a second possible metric: the number of common tones (or more precisely, the number of uncommon tones). The more tones two chords have in common, the "closer" they are considered. This works especially well if you plot chords as shapes in a Tonnetz grid. In this case, all of your triads are seen as triangles. The "closest" chords by this metric are those that share two common tones, which results in graphically "flipping" the triangle along one of its three edges. This would imply, for example, that the C major chord is equally close to C minor, E minor, and A minor (a single flip will always turn a major into a minor, and vice versa). In Neo-Riemannian theory, these types of transformations are even named: Parallel (P), Leading Tone (L), and Relative (R), respectively. There are more complex transformations between chords containing only a single common tone. Ignoring the 7th for simplicity, this metric would give your progression the following distances: 1, 2, 2, 2. This metric is less restricted by tonality, and more focused on voice leading. It can more easily explain the "closeness" of chords like C and A♭ which would traditionally be far-removed. As such, it is more suited to Romantic music, where these traditionally-distant progressions are more common. This metric also inherently accommodates different chord types.
There is an even more complex metric that has been developed by Dmitri Tymoczko in A Geometry of Music, involving n-dimensional orbifolds, but I cannot claim to be very familiar with it. It is well-suited to making you forget about music, and focus on mathematical abstractions.