Sounds like you basically want a static host for your blog posts and files. There are plenty of providers for this, I'd encourage you to look somewhere other than Dropbox though. They are a service for keeping documents on the cloud with access for specific users, not a static web host. On a slight tangent this reddit post about misusing Dropbox might interest you.
You can use the AJAX plugin to request files from a server easily enough. You should be aware that unless the server is set up with the correct "access-allow-origin" header you will not be able to request files from it if your web app is running on a different domain. For instance if you had web app on myblogapp.org and the files on blogstorage.org, then blogstorage.org will need to add the header "access-allow-origin: myblogapp.org" to all requests it serves. This can of course be solved by keeping the app and the data on the same domain.
GitHub pages is an example of a free hosting provider which could allow you to implement something like this, I don't think they offer configuration for HTTP headers so you would have to host the data and the app on the same domain.
Another option is AWS. The service itself is quite complex to find your way around, but last I checked there was a quick setup button on the AWS console homepage for creating a static web pages using S3, CloudFront and Route 53 ( storage buckets, CDN routing and domain name services ). On AWS you pay for usage, not a flat fee. For a low traffic site your probably looking at less than £1 a months for hosting and £8 a year for your domain name.
I've heard netlify is also quite good, and easy to set up. But I don't have any experience with them.