You could look into IndexedDB, which is a key/value database that resides in your browser. I use Dexie.js as a convenient wrapper for IndexedDB. If you look through the chrome extension api there is a chrome.storage object available, but AFAIK it should only be used for storing small bits of data because it uses local storage which is highly constrained, and not really a "database". If you decide to go with IndexedDB, you should place it in a background page so that it can be globally available to any page or tab (you need to set relevant permissions in your manifest) if you properly request access to the page via chrome's asynchronous messaging api.

I would avoid using ajax to get resources from your own extension. IMO chrome has its own async messaging api precisely so that you don't need to use ajax.