First convince yourself that an obsessive universal machine exists.
Then the following diagonalization argument shows that the set of non-obsessive machines cannot be recursively enumerable. Suppose machine $N$ halts exactly when the input is a non-obsessive Turing machine; then construct the following machine $D$ using standard quine/diagonalization techniques:
machine D is:
ignore any input;
construct Y as a description of D itself;
simulate N on input Y;
go to a distinguished penultimate state;
The construction of $Y$ is always the same, so it can easily be arranged to happen obsessively, simply by leaving out states that are not needed. Also, the simulation of $N$ can be done using an obsessive universal sub-machine. Thus the obsessiveness of $D$ depends solely on whether the distinguished penultimate state is ever reached. But if $N$ is correct, this will happen exactly when $D$ is not obsessive, so it is obsessive if and only if it is non-obsessive -- a contradiction. So $N$ cannot exist.
The set of obsessive machines cannot be recursively enumerable either. Using the obsessive universal machine, it is easy to translate any machine $P$ into one that is obsessive if and only if $P$ halts on all input. Thus enumerating the obsessive machines would lead to an enumeration of all Turing machines that compute total recursive functions, which is well known to be impossible.