If I refer to the status of something as enabled and disabled, the verb is to enable and to disable.
Are there comparable single words for the status of something being online and offline? So far I've got "put online" and "take offline".
"put online" and "take offline" are used for things, like servers or services.
"come online" and "go offline" are used to describe the action of a user using one of those services when he performs a login or logoff
So, sequentially, the service has to be put online, so that the users can come online inside that service. If the service is put offline, all users inside that service are forced to go offline.
Connected and disconnected.
As far as I know, to come online and to go offline are used in different online chat rooms/MMOs' ingame chats and other such systems.
Edit: Your own suggestion is another good variant, applicable if there is someone/something else that performs the operation. E.g. My friend went offline right in the middle of discussion vs. We had to take the server offline to perform maintenance on it.
In this day and age, if some resource is capable of being online at all, that is its 'normal' status.
The word offline does sometimes get used as a verb, as in "I need to offline the server".
I don't know of any similar usage for the opposite process (I've never come across "Online the server", for instance). Most people simply say "reconnect", or "put/get back online".
If the context is correct (i.e. it is established we are talking about something on the Internet) then you can say available and unavailable, although that's for state and not a verb, so you would end up saying make available and make unavailable which puts you in a similarly verbose position.
"online" is not a verb; you cannot "online" something. You can go online, but then "go" is the verb.
It might be the case that you could think of "to go online" as a phrasal verb -- I suspect that in the future, "to go online" will become an accepted phrasal verb, but I doubt that "online" itself will every become a verb.
From the users' perspective, a website is accessible when it is online, and inaccessible when it isn't.
EDIT: You're necessarily looking for one-word. One option is Activated / Deactivated.
Here's my two bits:
Log on and Log off
When you Log on, you are online, when you log off, you are offline