There's certain ambiguity in your question. You see, the implicit actions that the compiler takes with regard to constructors involve both declaring them and defining them. If some constructor is declared but not defined, do you consider it exists or not?
Anyway, there's no way to create a class that has no constructors declared for it. The copy constructor, for one example, is always declared. There's no way to suppress it. If you don't declare it yourself, the compiler will declare it for you.
As for the default constructor - it is possible to suppress its implicit declaration. If you declare any constructor yourself (i.e. explicitly), the compiler will not implicitly declare the default one. But in this case your class will, of course, have a constructor: the one that you declared yourself. (Plus, as I said above, the copy constructor is always declared).
As for implicitly defined constructors... They are defined by the compiler only if you use them. And, of course, they are defined only if it is possible. (If you use an implicit constructor, and it proves to be impossible to define, then your program simply will not compile).
So, once again, when it comes to declared constructors, it is not possible to have a class with no constructors at all. Any class has at least one constructor declared for it.
If your are interested in defined constructors specifically, then it is indeed possible to have a class, for which no constructor is defined. Here's an example for you
struct S {
S(const S&);
};
That's it. The class has one constructor declared it in, but it is not defined :)