Under what models do we know linear time sorting? The best we know for general case sorting is $O(n\log n)$ (which is also $\theta(n\log n)$ is decision tree model) and the problem of $O(n)$ sorting is open for turing machine models.

Under what models do we know $O(n)$ algorithms and how far they are deemed away from general models?
One obvious answer is "spaghetti sort", or in other words - sorting in a spaghetti model.

Intuitively, the spaghetti model says that your integers are given as lengths of (uncooked) spaghetti, and you sort them by placing them on a table, and then lowering your hand until you hit the tallest spaghetto (for spaghetto is the singular of spaghetti). You take that one out, and repeat.

A bit more formally, this model gives you the operation $\arg\max\{a_1,\ldots, a_n\}$ in $O(1)$.

Determining how "far" this is from a general model is a vague question. There is a trivial lower bound of $\Omega(n)$ on computing the maximum of $n$ elements in a RAM model (and an even worse bound of $\Omega(n\log n)$ in a single-tape Turing Machine), so this is not a realistic model. 
You could say that it's equivalent to a RAM model with oracle access to $\max$. I'm not sure how interesting that is.

Another cute aspect of this model is to consider it's physical properties, in the intuitive spaghetti description: recall that in this description, you lower your hand onto the spaghetti. Now, if you actually want to do this to an arbitrary number of spaghetti, then you would need an increasingly large hand. 
So this model induces a notion of "hand size" complexity.