Use the GFCI loophole
There's a rule (and for very good reason) that if its two supply wires have GFCI protection already on them, you can fit a 3-prong receptacle there even though it doesn't have a proper ground. GFCI prevents electrical shocks by detecting ground faults; power taking a route it should not. This is not quite the same as a full and proper ground; so for instance if you touch a metal chassis and get a spark of static electricity, that has nowhere to go.
Receptacles are typically wired in a string, like the larger screw-base Christmas tree lights. The 2-wire cable goes from the service panel, to the first receptacle, to the second, third and so on. It's important to know which receptacle is first on that chain.
You can fit a GFCI receptacle at that first location. The rest of the chain can be attached to its LOAD terminals, giving them GFCI protection. They will need stickers saying "GFCI protected". Then you can change them to 3-prong outlets. They cost about $3 for a decent one (avoid the 60 cent cheapies). Then you'll need to stick "no equipment ground" labels on all those receptacles, and "GFCI protected" labels wherever that's not obvious.
If you don't want to sort out which one is first, just change the circuit breaker to a GFCI breaker. These are more expensive, and they also prevent you from fitting a simple AFCI breaker. AFCI detects arcing and sparking from wiring faults; left alone they will start a fire.
Use the ground-retrofit rule
The 2014 electrical code threw open the door to retrofitting ground wires. It is debatable whether grounds are as good as GFCI for life safety... but they complement GFCI's very well. And of course they help protect equipment from static and lightning.
The new rules allow you to leave the existing wiring alone and simply retrofit a ground wire using any practical means. You don't need to run a ground all the way back to the panel, or even follow the same circuit paths as the wiring. You only need to reach any existing ground wire, provided that wire is thick enough and goes back to the same panel. So for instance you can hang off a range or water heater cable, or the one going to that nice NEMA 6-20 receptacle in your photo.