Use a household upright vacuum cleaner, not the nozzle type like at the car wash. Lay the carpeting on a flat clean surface and go over it for several minutes with the vacuum. Hang them up on a clothes-line to air out in the sunshine. If they smell, a little baking powder sprinkled generously on them will also soak up most of the odors. Of course, vacuum the powder off before putting back in the car.
Floor mats come in many varieties.
My XJ-S had a Jaguar coco mat, but they could be carpet, rubber, or soft pile. You can wash and hose down rubber mats. Both the mats and carpets may have stains on them. If you want to really clean these well, then rent a professional steam cleaning system. I have found these to be safe to the carpet color and texture, and they do a good job of pulling out even more unbelievable amounts of grey-gunky stuff.