L = ϵ is nothing but Null string if we draw the DFA the first state is final

It is Perfectly Regular and since it i Regular according to chomsky hierarchy it is CFL,CSL,Recursive

TM accepts ϵ

Ref: http://math.stackexchange.com/questions/668896/what-does-it-mean-for-a-turing-machine-m-to-accept-epsilon

TM is nothing but Finite State Machine with 2 extra stack

If Finite State Machine + 1 Stack =PDA

Finite State Machine +2 stack =TM

So it is perfectly alright to say it is regular and since we can built a DFA so obviously yes TM can be build to accept ϵ