There are several ways to solve this problem.
Web based terminal
A web based terminal allows you to connect to your server via SSH though a web browser. These clients are usually AJAX based and require a connection to the back end server or even need to be hosted on the system itself. A big advantage with this method is that initial and repeated access is very easy, but it takes more time to initially set up. You would have to use a webserver to serve the page, secure it and make port forwards for the webserver itself.
More information, including specific products can be found on Wikipedia: link
Screen sharing solutions
If you run X on your local machine you can look into screen sharing solutions like TeamViewer or YuuGuu (which also work on Linux). Another category, without relying on a third party is VNC-like tools, but you would have to make a port forward on your router.
A special note, even though an Oracle account is required to use it, is the Oracle Shared Shell. It is a Java applet which runs from the browser and lets you set up an SSH connection to a machine on your network. Another user can then login to the session (like a video conference) and you can share the session. Originally designed to let Oracle support employees work on customers' machines, it does not require an Oracle employee to participate in the session.