|The Capella Santa of San Francesco della Vigna|
But cloistered as the above-pictured chapel is by, well, on one side quite literally a cloister, on another by a church, and on its two others by the structures of the monastery of San Francesco della Vigna, it's perfectly positioned at the northern edges of Castello to be among the quietest interiors in the city.
While the beautiful church of San Francesco della Vigna is often during its official visiting hours polluted with the noise of recorded classical music--since when do churches and great art require an obnoxious piped-in soundtrack?--the Capella Santa, separated from the church by a broad hallway and closed doors, is always blissfully quiet, and usually unoccupied.
I've also sat looking at the late Giovanni Bellini painting in there, trying to figure out which parts of it were done by the master and which by his workshop. A full account of this painting and the church as a whole can be found on the excellent website The Churches of Venice: http://www.churchesofvenice.co.uk/castello.htm#sanfrandellavig
Perhaps it's not among the greatest of Gio Bellini's works, and as you'll find on the above link, Giorgio Vasari claimed it was painted mostly by one of his pupils, and yet there are elements (like the gaze of San Sebastiano out upon you) that won't quite let you go, and its hushed composure makes the chapel seem all the more intensely quiet.