By definition, if the site doesn't respond when you ssh into it, it doesn't accept ssh connections from you. But that could be due to a temporary failure or a problem at your end.
ssh -vvv tigershark.engr.sjsu.edu to see where the process hangs.
If it responds to pings, it means that there is a machine with that IP address that is presently connected to the Internet and that responds to pings. The machine may however not be running an ssh server, or your login attempt may be blocked by a firewall. The firewall could be anywhere from your machine (unlikely), through your network (some sites block outgoing ssh connections), to the target network. If you can ssh to
tigershark.engr.sjsu.edu from within SJSU but not from outside, that's the likely explanation. Maybe there's another machine you're supposed to connect through when you're off-campus; ask the system administrators.
tcptraceroute tigershark.engr.sjsu.edu 22 will try to send packets to the target port and report how far the packets went. You may need to install it via your OS's package management system or from the tcptraceroute website.
A very powerful tool to see what services are available on a network is nmap. Beware that depending on what options you use and who you ask, the level of intrusiveness may range from phoning a store to ask the opening hours to checking if the backyard-facing window can be open from the outside by inserting a knife blade just there and twisting just so.