I know git supports SSH. But my 13-year old server does not support SSH. Only telnet is supported.
Is it possible to push to a git repository over telnet?
How to setup such environment?
No, it isn't possible.
Even 13 years ago telnet wasn't an acceptable protocol to use, and ssh was already supported by any server worth using. That's even more the case now.
You may be able to set up support for doing git over http, including pushes. But you should really see about getting something that can support ssh.