Is it possible to access an SQLite database from JavaScript?

You could use SQL.js which is the SQLlite lib compiled to JavaScript and store the database in the local storage introduced in HTML5.

Actually the answer is yes. Here is an example how you can do this: http://html5doctor.com/introducing-web-sql-databases/

The bad thing is that it's with very limited support by the browsers.

More information here HTML5 IndexedDB, Web SQL Database and browser wars

PS: As @Christoph said Web SQL is no longer in active maintenance and the Web Applications Working Group does not intend to maintain it further so look here https://developer.mozilla.org/en-US/docs/IndexedDB.

SQL.js

EDIT

As @clentfort said, you can access SQLite database with client-side JavaScript by using SQL.js.