Is there a way to change the width and height of closed captions?

If you use the modern player you have some control over caption appearance (font size, colours, position) in Player Properties | Colours & Effects.  Hopefully that should be enough for you.

Otherwise for more control you have to resort to adding some CSS to your published story.html & index_lms.html files.  For example this:

<style>
div.caption-container > div.caption {
   left: 20% !important;
   font-size: 75%;
}
</style>

will make the text tiny & move the box towards the bottom left.

The best way to work out what CSS selectors & properties you need to adjust is to run the published file, pause a captioned video and then use your browser's developer tools to target the various CC related elements.  Then tweak & fiddle with the CSS properties until the captions look like you want.

Here's what the process looks like in Firefox dev tools:

You'd have to use an app like Premiere Pro for this.