@Sea 

Wow, that is some complex wire colours! Usually home electrical cabling are Black (hot), White (neutral) and Green (ground). Sometimes there is also a Red which is either a second hot circuit coming from the panel, or used between two 3-way switches to add another Traveler.

A couple of observations from your pics. If the green cables bundled together in the Marrette at the back are also connected to the box screw in the back left (looks like it), then those cables are Ground (which is good from a safety point of view, and good use of colour).

The downside is those cables are not neutral. Usually if there are neutral cables in a box, they too are bundled together in a Marrette.Certainly none of the cables connected to the switches are neutral. Some of the cables that seem to loop between one entry and another could be neutrals, but there's no slack to access them and troubleshoot what they are.

Regarding depth, those cylindrical entry posts are quite big and likely in the way of anything deep. And from my experience, all smart switches from various brands are deep. There are some smart light switches and dimmers that do not require a neutral wire, but I think they are still too deep to go in that box. So it doesn't look like you can put a smart switch in there.

How old is your house?

Cheers,

Rick