Enumerating the set of polynomial time Turing machines
The halting problem reduces to $\overline{L}$: given a machine $T$, create a new machine that on input $n$ simulates $T$ for $n$ steps, and if $T$ halts then it enters an infinite loop. (I leave you the details.)

Since the halting problem is complete for $\mathsf{RE}$, it cannot be the case that $\overline{L} \in \mathsf{coRE}$, and so it cannot be the case that $L \in \mathsf{RE}$.