Is it possible (and safe) to make an accepting socket non-blocking?

No idea about Windows, but the behavior you want is guaranteed by POSIX:

If the listen queue is empty of connection requests and O_NONBLOCK is not set on the file descriptor for the socket, accept() shall block until a connection is present. If the listen() queue is empty of connection requests and O_NONBLOCK is set on the file descriptor for the socket, accept() shall fail and set errno to [EAGAIN] or [EWOULDBLOCK].

Source: http://pubs.opengroup.org/onlinepubs/9699919799/functions/accept.html

Also, select or poll can be used to check for incoming connections by polling for the listening socket in the reading set.

In the question, You are saying that you do not want to use select (or poll or epoll) which are the best ways for IO multiplexing. I would recommend you using one another thread just for listening sockets while this is a bad idea!