Well, some of it is going to depend on what you consider important for a virtual pet game. I can't think of a way to have time pass when you don't have the game running, for example.
However, there's no reason that you couldn't make something a little like that. You'd create a lot of parallel process events that would probably use the timer or very long wait commands to periodically change the pet's assorted conditions, and other events that use keyboard input to do things like feed the pet, which would reset the "hunger" track. Scripts could probably make this easier, of course, but they'll be doing essentially the same thing. (If a scripter comes along and offers a better solution, please listen to them. I am not a good scripter.)
It would, of course, be complex, and you'll need to do a lot of work, but I wish you luck!