Bicycling 36 km per day in city traffic is feasible?
I have a co-worker who rides into work on a ~30 mile one way trip. It's no problem for him because he's a good cyclist, has a good bike, and is used to that kind of distance. He's got his saddle worked out, his form is good, his lights are good, and he knows his route. 

The biggest problem he has is dealing with a locker room, and needing to move clothes back and forth. He usually drives in on Friday to haul his used up work clothes back his house. 

The things I would focus on:


The logistics of clothes, weather, being clean, and your schedule.
Knowing how to fix a flat.
Having a back up plan when you wake up and it's raining/snowing/hailing outside.
Getting used to the mileage. If you've never really ridden before you'll be dealing with soreness, kinks, and gear issues. 
Good bike maintenance. If you keep everything lubed up, fix little issues before they get big, and can do your own work (or most of it) you'll have a lot less problems.


I was riding my bike your distance to work when I was 14 years old, but I was also a competitive cyclist so the distance was nothing. A mid-range competitive cyclist can easily pack away 400 mile weeks in the winter so your co-workers are nuts that it's not possible given you have the body and bike to do it.