Semantically-Accurate HTML5 Element for a Modal Dialog

The current draft of HTML has a dialog element. It was removed at one point but it's back, and browsers are starting to implement it.

The dialog element represents a part of an application that a user interacts with to perform a task, for example a dialog box, inspector, or window.

For accessibility (until the element is more widely supported) I would also include the dialog ARIA role.

To get the JavaScript API in browsers that don't natively support the <dialog> element, you can use a polyfill.

<aside> seems appropriate. The current spec with relevant sections bolded:

The aside element represents a section of a page that consists of content that is tangentially related to the content around the aside element, and which could be considered separate from that content. Such sections are often represented as sidebars in printed typography.

The element can be used for typographical effects like pull quotes or sidebars, for advertising, for groups of nav elements, and for other content that is considered separate from the main content of the page.

In this case, a modal is "tangentially related" to the action that caused it. While you might normally expect an aside to be in a sidebar, one of the purposes of semantic content is to enable versatility that's unrestricted by physical page characteristics. The last phrase of the spec seems to imply just this versatile use case.