Can I obtain C++ type names in a constexpr way?

Well, you could, sort of, but probably not quite portable:

struct string_view
{
    char const* data;
    std::size_t size;
};

inline std::ostream& operator<<(std::ostream& o, string_view const& s)
{
    return o.write(s.data, s.size);
}

template<class T>
constexpr string_view get_name()
{
    char const* p = __PRETTY_FUNCTION__;
    while (*p++ != '=');
    for (; *p == ' '; ++p);
    char const* p2 = p;
    int count = 1;
    for (;;++p2)
    {
        switch (*p2)
        {
        case '[':
            ++count;
            break;
        case ']':
            --count;
            if (!count)
                return {p, std::size_t(p2 - p)};
        }
    }
    return {};
}

And you can define your desired type_name_length as:

template <typename T>
constexpr auto type_name_length = get_name<T>().size;

DEMO (works for clang & g++)

_{Edit: Updated based on this answer to the non-constexpr-specific question; it is the result of refinements by several people including @HowardHinnant, @康桓瑋 @Val and myself.}

The language standard does not - to my knowledge - provide any facility for obtaining type names. So, we resort to compiler-specific approaches. This works with GCC, clang and MSVC.

#include <string_view>
// If you can't use C++17's standard library, you'll need to use the GSL 
// string_view or implement your own struct (which would not be very difficult,
// since we only need a few methods here)

template <typename T> constexpr std::string_view type_name();

template <>
constexpr std::string_view type_name<void>()
{ return "void"; }

namespace detail {

using type_name_prober = void;

template <typename T>
constexpr std::string_view wrapped_type_name() 
{
#ifdef __clang__
    return __PRETTY_FUNCTION__;
#elif defined(__GNUC__)
    return __PRETTY_FUNCTION__;
#elif defined(_MSC_VER)
    return __FUNCSIG__;
#else
#error "Unsupported compiler"
#endif
}

constexpr std::size_t wrapped_type_name_prefix_length() { 
    return wrapped_type_name<type_name_prober>().find(type_name<type_name_prober>()); 
}

constexpr std::size_t wrapped_type_name_suffix_length() { 
    return wrapped_type_name<type_name_prober>().length() 
        - wrapped_type_name_prefix_length() 
        - type_name<type_name_prober>().length();
}

} // namespace detail

template <typename T>
constexpr std::string_view type_name() {
    constexpr auto wrapped_name = detail::wrapped_type_name<T>();
    constexpr auto prefix_length = detail::wrapped_type_name_prefix_length();
    constexpr auto suffix_length = detail::wrapped_type_name_suffix_length();
    constexpr auto type_name_length = wrapped_name.length() - prefix_length - suffix_length;
    return wrapped_name.substr(prefix_length, type_name_length);
}