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Based on this log file, one of the following is going on:
1. You are accessing your Akeeba Backup Control Panel page while a backup is being taken. DON'T do that.
2. You tried to start a new backup while the previous backup had not completed yet.
3. Another backup is already running when you try to take a new backup.

In order to fix the latter issue, I would suggest the following:
- Please make sure that you are not using a scheduled backup, e.g. using CRON on your server or webcron.org
- Make sure that you are not using the "System - Akeeba Backup Lazy Scheduling" plugin which was deprecated since Akeeba Backup 3.2.5, discontinued since 3.3.a1 and is known to screw up your site since 3.3.5 (and we do try to forcibly uninstall it when you install Akeeba Backup 3.3.5 or later)
- If the above won't help, try removing all files from your backup output directory. The default location is administrator/components/com_akeeba/backup but you may have overridden this in the configuration.

If the above don't help, please tell me the exact error message you see on your screen (just telling "AJAX error" doesn't help, I need the full error message) and ZIP and attach your log file so that I can help.

OK, this is a more sane log file and your explanation gave us the solution. Your PC going to sleep killed the backup.

When your PC goes into sleep mode, it actually turns itself off which means that the network connection is cut off. This is equivalent to closing the Backup Now page. Remember the warning on the top of that page which tells you not to close or navigate away from this page while the backup is in progress? Essentially, your PC did what the warning on that page explicitly told you not to do.

And I will further explain this. In order for the backup to not time out, we have to split it to smaller chunks called "steps". However, your server can only run one step at a time. We have to somehow tell it to go on with the next step, and the one after that and so on until the backup is complete. How can we do that? By having your browser send your server some commands through AJAX. Essentially, your browser tells your server to go on and your server replies to your browser how far it's into the backup. When the browser-server connection is severed in mid-process, the browser has no idea what the heck is going on and throws an AJAX error.

So, just tell your computer NOT to go into sleep mode while the backup is running and make sure that your Internet connection does not disconnect during backup.

You're welcome!

I would suggest that you read our Automating Your Backup section of the documentation. The easiest, fool-proof method of backup automation is using the instructions on that page with webcron.org. Since it's a web service, you don't even need to have your PC turned on; it will backup your site for you and will only cost you about 3$ / month for daily backups or even less if you only plan on doing weekly backups. In fact, all you'll have to do is turn on your PC in the morning and let DropBox sync the new backup archives. Or you could try creating a CRON job on your own server and not have to pay anything. IMHO, spending a couple of bucks per month to have peace of mind is worth it, so I'd go with webcron.org any time :)