Is it possible to move an item out of a std::set?

No, it is not possible. There is no way to get non-const access to elements in a set, and move requires non-const references. Allowing non-const access would make it trivially easy to break the invariants for set.

C++17 added a function std::set<>::extract that allows moving an object out of a set:

std::set<MoveOnlyType> s;
s.emplace(arg0, arg1, arg2); // only way to insert such move-only objects, since C++11
auto internal_node = s.extract(s.begin()); // internal_node no longer part of set, we can do with it what we want
MoveOnlyType m = std::move(internal_node.value()); // finally get the actual object out