Vented or not?

I'm thinking May Be, if the down pipe ties into a vented line.

Is that possible or this is just plain out of code?

I think it depends on how long the horizontal run to the vented main is.  I don't recall the exact dimension, but I believe this is called the 'weir', and the permitted length is determined by the diameter of the horizontal pipe.

Then again, I think what you have is a 'full S trap' and that they went out of code some years ago.

Red is right that is a full S trap and it is definitely no longer (probably in the last 75 years) allowed by code. 

This is a trap that can siphon itself and allow bad stuff into your home.