Electrical – a safe way to ground this light fixture

Here's my best guess at the wiring based on the photo. Looks like you've got one hot/neutral coming into the box from the panel (or another switch/outlet), and you've got 2 hot/neutrals going out of the box to other switches/outlets. Since these are all joined together, it doesn't really matter which one is coming in and which two are going out.

Then it looks you you have a white wire connected to the bundle of hot wires, and that white is going out of the box to the switch, and then a black wire returning from the switch, which goes to your lamp. This is pretty common, but that white wire should probably be marked at both ends to indicate that it's actually a hot.

Based on what I can see in the picture, you do not have a ground wire. It's possible that the box itself is grounded, either with a ground wire screwed to it that's not visible in the photo or by metal conduit, but based on the yellowed insulation of the white wires, my guess is that your house is old enough that no ground was required at the time, so there is not ground wire.

Update based on your close-up picture, there are ground wires, they are sort of terminated to the box on the strain relief clamps. If you have enough slack, it would be good to wirenut them all together securely with one pigtail that goes to the box and one to your lamp. Otherwise, make sure they are all securely fastened to the box, only one of them (the one that leads back to the panel) is a "real" ground, the others are providing ground for downstream lights/outlets, so make sure that all all them are securely fastened.

In a single circuit, all ground wires and metal fixture structural parts are connected.

In your case, the green wire is the ground for the lower part of the fixture. The bare wire is the ground for the metal canopy. The green screw is the ground for the fixture bracket. The bare wire buried under the bracket is the ground from the circuit cable.

They all need to be connected. The easiest way is to attach a short bare (or green) wire to the green screw and then attach the four ground wires together.

I prefer the newer push in connectors for solid wires (they can be used for stranded but are a bit harder to use). Or you can use a wire nut. Just be sure to strip enough bare cable to get a good twist on all four wires before putting on an appropriately sized nut. Most practitioners use a lineman's plier to get a good twist on multiple wires.

Never put more than one wire under a screw or nut.