The line you are (were) on or off of, was the current loop. This is 1960's/1970's (Teletype Corp ASR-33, but the line goes back to the earliest days of telegraphy wires strung long distances outside).
"ON LINE" means your teleprinter is wired in series with a constant-current loop; timed interruptions of the current signalled characters which the printer printed. The keyboard, wired in series with the printer, could induce characters.
These were half-duplex; at any moment, send or receive, not both. Or, typing, it printed locally and remotely. One pair of wires, 60 milliamperes at 100 volts (or so) comprised the loop. (The ASR-33 usually jumpered for a mode modern 20 mA).
