'Scuse me, but I was involved in this kind of thing professionally. You can have a bench mat and your work pieces on the bench mat and your wrist strap connected to the bench mat, but there is no need to connect the whole lot to earth. All the official instructions say you do, but that is an extra level of safety and not strictly necessary purely from the static control point of view.
For working on equipment when there are no other static control measures: if you are yourself insulated from earth (and with modern fabrics, rubber soled shoes, etc you probably are), then touching the equipment chassis will be sufficient to bring yourself to the same potential as the equipment without needing to worry whether there is an earth connection. Do not however move much without re-grounding yourself, because man-made materials have a habit of generating static when rubbed.