Sometimes my machine tells me that a certain action can only been taken by the 'administator'. I am a one-user machine on Win98SE on the Internet (no LAN). How do I get myself appointed as 'administrator'?

   Some Windows installations set up an extra account which is the administrator account.  If that is your case, you need to log in to that account to have administrator privileges.
   If you go to User Accounts ( it may be called something else in Win98 ) in the Control Panel, it will show you who the administrator is.

I find that when I try to open 'Users' in Control Panel the machine tells me the 'administrator' has disabled it! Can I get 'round this by tweaking regedit in some way?

   No, not that I know of.  It is a security issue and so I doubt that they would make it that easy.
   You will have to log in on the adminstrator account.  Since you don't know what that is, there is not anything that you can do but live with it or delete everything and reinstall Windows.
   Maybe someone else has a better idea.

If you can run regedit, but cannot open and view the users accounts then someone made some odd security settings on the computer.
Do you have to log on with a password to the computer or will it boot to the desktop from the log on screen by hitting esc key?
Was this computer previously owned by a company or some unknown person?
Best solution, as stated by Ray, may be to back up any personal data, then format hard drive and reinstall Windows.

No, the machine was not previously owned and nobody has access to it bar myself so whatever it is is down to me. I also have the usual security precautions in place (including ZoneAlarm and AVG and HackThis) and diagnostics show all my ports running in stealth even though I'm running PWS and a large website on a virtual server.

Yes, I can use regedit freely and most other things too. It's just ocassionally I find that 'Mr Administrator' pokes his head in!

I'm a bit curious as to what's telling you that you need to be administrator since Win98SE has no privileges.