Is it possible to use std::string in a constexpr?

As of C++20, yes.

As of C++17, you can use string_view:

constexpr std::string_view sv = "hello, world";

A string_view is a string-like object that acts as an immutable, non-owning reference to any sequence of char objects.

No, and your compiler already gave you a comprehensive explanation.

But you could do this:

constexpr char constString[] = "constString";

At runtime, this can be used to construct a std::string when needed.

C++20 will add constexpr strings and vectors

The following proposal has been accepted apparently: http://www.open-std.org/jtc1/sc22/wg21/docs/papers/2018/p0980r0.pdf and it adds constructors such as:

// 20.3.2.2, construct/copy/destroy
constexpr
basic_string() noexcept(noexcept(Allocator())) : basic_string(Allocator()) { }
constexpr
explicit basic_string(const Allocator& a) noexcept;
constexpr
basic_string(const basic_string& str);
constexpr
basic_string(basic_string&& str) noexcept;

in addition to constexpr versions of all / most methods.

There is no support as of GCC 9.1.0, the following fails to compile:

#include <string>

int main() {
    constexpr std::string s("abc");
}

with:

g++-9 -std=c++2a main.cpp

with error:

error: the type ‘const string’ {aka ‘const std::__cxx11::basic_string<char>’} of ‘constexpr’ variable ‘s’ is not literal

std::vector discussed at: Cannot create constexpr std::vector

Tested in Ubuntu 19.04.