Is it possible to set up a forum with a system of invitation only? By this I mean like Gmail or Orkut where there is no signup page unless you have been invited.

Yes, it is possible.
You may use any forum of your choice and modify it suit your needs.

as far as I know you could make registrations in phpbb2 be activated by administrator,therefore you could register the users you only like

That is correct. In admin panel select configuration (under general admin) and set "Enable account activation" to Admin.

However this doesn't work quite like Gmail for example. As anyone can try to register and admins can only approve their accounts.

I have follewed the phpBB mod "scene" quite much and haven't ever seen a mod that would enable invitation system exactly like Gmail's is. There might be such available for other BB systems as vBulletin (maybe new versions come with such option already) and Invision.

Thanks for your replies!

That's not quite what I wanted, this is slow and isn't really the way gmail works. An automatic generated link for a signup registration that lets you use it once to create one account is what I was trying to find. I am awful with all kinds of coding, I was just wondering if there is anywhere for Invision to do this?

You can always get robots not to find your forum by having no links to it from any sites then giving to link to people you want to invite

u can do this in most forums i think, just fiddle around with registration or validations stuff or something in the admin panel.

btw, gmail is open to anyone now, u dont need to be invited.