GoDaddy, like many other shared hosts, is trying to cram as many sites as possible on a single physical server. A server has a finite amount of resources (processing power, memory, hard disk space, network bandwidth). The only way to cram as many sites as possible is applying restrictions to how much resources they can use.
GoDaddy used to have very tight resource limits but it was possible to make Akeeba Backup's backup process and Admin Tools' PHP File Scanner process run 3-5 times slower than normal to work around it. Since July 18th, 2014 the resource limits on GoDaddy became unrealistic. You'd need to make the backup 30-100 times slower. A backup of a relatively small site which normally completes in two minutes would need somewhere between one and four hours. A backup of a large site that normally needs an hour would now take days.
The limits GoDaddy has chosen are UNREALISTIC. They only demonstrate that their servers are oversold and used well beyond their capacity. They are effectively telling you that they charge you as much as other companies, or even more, for a service that's far worse and for servers that cannot cope with the demand! The only course of action you can possibly take is first notify GoDaddy about the problem they have caused, then move to a decent host if they do not want to change their server settings to be more realistic. We strongly recommend SiteGround and Rochen for hosting your Joomla! or WordPress site.
Unfortunately, we cannot do anything on our part to help you. We could give you settings which would make the backup run desperately slow –taking days to back up a four hundred megabytes site– but this is impractical and nothing sort of a joke. For the record, a 400Mb site on Rochen hosting backs up in two minutes thirty seconds. Rochen charges pretty much the same as GoDaddy. That's the situation, folks. The only course of action on our part is to decline providing support to GoDaddy clients because their issues are caused by GoDaddy settings which are outside our control. Any tickets regarding issues on GoDaddy will be sent a link to this page and closed.
Important clarification: host-related issues are not a valid reason for a refund per our Terms of Service. The reasoning behind this is that it's easy to cancel your hosting account with the bad host (and get a refund of the unused hosting time if you paid in advance) and move to another, decent host with pretty much the same monthly cost and a much better service level.
Thank you for your understanding.
Update July 23rd, 2014. We are already in contact with GoDaddy in order to resolve this issue. We have given their engineers all the details we have collected so far and they are trying to reproduce the issue and isolate the setting which triggers it. We will post further information as we are receiving it.
UPDATE July 24th: GoDaddy engineers have identified and the issue and are working on it. When the problem is fully resolved we will resume support to GoDaddy clients. We don't want you to think that we are making stuff up, so here is the reply we got a few minutes ago from the GoDaddy engineer: "We think we found the issue in an infrastructure update made around the time you saw this start to fail. Your timing and sample accounts helped us track it down. It also impacted an image upload tool that the Managed WordPress team was using to load images in bulk that had a similar AJAX pattern. We're working on rolling back that patch this morning and then will test then will let you know."
UPDATE July 29th: GoDaddy engineers have identified the issue and have deployed a reversal of the patch causing the backups problems since July 26th. However, it takes a while for all of their servers to become up to date. We have tentatively removed the warning about not providing support for GoDaddy customers but please DO NOT request support for backup failures on GoDaddy until August 4th as we can't know if your server is affected or not.