The mechanics i knew/know didn't care if they were greasey like the picture and would eat food while working under or in a vehicle. 1 guy cared a little, he was mostly a diesel mechanic, his greased up hands he would "wash" his hands for lunch, but his sink was a 55gal drum of diesel fuel, he just put his hands in did a quick clean-off and take the food and start eating it. And unless or until you work on a car, get hungry and a PREC (how most people have them) was coming your way, the hand cleaning thing doesn't really happen or is quick.... hahaha...
42 years as a mechanic and I can absolutely swear this is true.
As for eating, being a city mechanic, we had to fix the garbage trucks. After a while, you just get used to it. But we did wash our hands before eating. Its always fun to watch the new guys the first time they have to climb into one of those stinking things.