As far as I know the get_avatar() function only allows for square values. That doesn’t mean you couldn’t use styling to display a rectangular avatar. Essentially you would use styling to “shave off” 150px from the width.
So, let’s assume that your theme file produces a 300 x 300 pixel avatar for the post author using the following code:
<?php echo get_avatar( get_the_author_meta( 'ID' ), 300 ); ?>
It’s then only matter of enclosing the avatar image inside of a DIV tag that you can style, like so:
<div class="auth-avatar">
<?php echo get_avatar( get_the_author_meta( 'ID' ), 300 ); ?>
</div>
You then use CSS to crop and center the image:
.auth-avatar {
width: 300px;
height: 150px;
overflow: hidden;
}
.auth-avatar img {
width: 100%
}
That should do the trick. Here is a fiddle with the results: