It says that I must offer "an opportunity to receive the Corresponding
Source of [my] version by providing access to the Corresponding Source
from a network server at no charge".
There's no indication that I can delay this opportunity at will, as in
"yes, to get source click here, but maybe you have to come back
tomorrow".
If you setup a system which required a delay, that would be questionable. For example, there are commercial download services which permit free downloads after a delay (say, 5 minutes) to encourage paid membership for instant downloads. Whether this is ok with the (A)GPL can be debated.
However, if the source is temporarily unavailable not by your intention or fault, then so long as you make a reasonable attempt to make it available (ie, somebody emails you to let you know the source server has been down, it doesn't come right back up, and you upload it to another server) you're still in compliance.
The word "continuously" at no point appears. 99.999% uptime is not demanded by the license. It is expected that things fail on the Internet.
It seems I am obligated to ensure the Corresponding Source is available
as long as I offer access to Object Code...
"As long as" represents duration of offer, not continuous/simultaneous.
As long as you're still offering the binaries, you must still offer the source code. This does not mean if the server hosting the source code goes down, you must also take down the server hosting the object code, so long as you get the source code server back up in a reasonable amount of time. Shit happens, it's expected, and the (A)GPL has no teeth to bite anyone in the ass just because a server fails.
Heck, if you mistakenly fail to comply with the GPL's source offering requirement, and someone informs you of it, you have 30 days to correct the problem while you're still able to distribute the object code in full compliance with the GPLv3.