There's nothing you can do about it. The site which provides the web space for the blog does so on the basis that the blog only receives light usage. If the site gets too many 'hits', users are automatically locked out of the site.
Most providers of web space monitor site usage on a daily basis and lock the site out of use as soon as the daily traffic limit is reached. If so, they normally 'unlock' the site at the start of a new day. (The actual time varies. It's usually midnight at the location where the server is located. So, if the blog is hosted on a server on the East Coast of the USA, a new usage day will start at midnight EST, which is 0500 BST). So there's a good chance that you'll be able to access the blog as soon as a new usage day starts.
However, some web space providers monitor traffic over a longer period. If it's excessive compared with the amount that the site owner is allowed, they'll only unlock the site (= blog, in this case) if the site owner/blogger agrees to sign up to a premium deal (which involves the payment of a fee). Alternatively, of course, the blogger can simply relocate the blog to a more generous provider. Under such circumstances, the blog might be unavailable for some time (until the matter is resolved) or simply disappear forever.
Chris