You can access the Galileo's console through the serial port (the one that looks like an audio jack). However, you will need the proper cable to access the console. Take a look at the following document, the cable you need is showed there: Console Serial Cables for Boards and Kits
If you want to access the console using the Ethernet cable, you have to use Telnet. You have to start the telnet server on Galileo and assign the IP address to get access. Please check the following thread for additional details: Using Galileo Linux over Ethernet(Telnet).
ssh is only available in the SD Linux image. There is no ssh in the "small" image located in the serial Flash. So, if you need absolutely ssh, then you must burn the full Yocto image on a SD card and boot from it.
By the way, I *highly* recommend you to use the Galileo in that way, the capabilities are much, much better (the first one being the Galileo being able to retain its sketch when powered off )