I'm using a Raspberry Pi 3 running Raspbian. All I have available is an RS-232 port through an USB adapter to my PC running Ubuntu 16.04.
Is it possible to establish an SSH connection using the RS-232 wire?
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You could, but it would be a fair bit of work to set up – so for initial setup it probably isn't an option unless you put together a custom image. Usually, ssh is used to secure a network login session. Over a local serial (RS-232) connection all you need is your USB serial cable (wired as a "null modem" cable) and your favorite terminal emulator.
However if you want/need a network between the Pi and the Linux machine you could set it up to run PPP over the serial RS-232 link. Then you could run ssh or any other network traffic across the link.
No - by that I mean it's impractical and there is little or no "value" in doing so. It's a dedicated, physical connection, and therefore far more secure than SSH.
What you seem to want is to run a console over a serial port, and "talk" to your Rpi3 without a network connection. If that's what you want, then you can establish communications between your RPi3 and your Ubuntu box. You state that you have a "USB adapter", but don't say how you plan on making the physical connection to the GPIO pins in the RPi3. I'll assume you're using a cable like this one from the pimoroni shop. Once you get it wired up properly, you'll need to make sure that the file on your microSD card "/boot/config.txt" does not contain the line: "enable_uart=0". If it does, then delete that line.