I've seen some very mixed results while searching on the internet of whether attaching documents to a blog post is a built-in feature for SharePoint 2013 and I'm not sure what to believe. Most people say it's not built-in but that might be for previous versions of SharePoint. Somewhere I read there should be an option in List Settings -> Advanced Settings to enable attachments on the Posts list but I do not see this and can't find any sort of built-in feature.
Attachments are technically available on any SharePoint list. Depending on the definition of the list, they can sometimes be obscured.
With the blog posts list, they are enabled (just not visible from the settings). If you browse to the list item view of the Posts list and select the checkmark for a post, the Attach File option is available on the ribbon under Items.
I agree, on 2013 i am not seeing the option through the UI. If you go through the API, you might can change the EnableAttachments property to true and see what happens. Most likely, there is a good reason why they have it turned off.
But with the new 2013 features of being able to easily insert images and upload files into the html of the body, i don't really see why you need attachments.
I agree with both John and Steve. Attachments can be done this way BUT there is a caveat to this approach. If you decide to remove the file link from the post after uploading the file, the file will not be automatically deleted from the library where it was uploaded. You will have make sure you manually delete that specific file otherwise it will stay there for the rest of its life as an "orphaned" item!