Overview
The classic block brings back the old editing features of WordPress' classic editor, also known as TinyMCE.
The only available options are the ones you see in the editing toolbar. There are no additional options; not even an additional CSS class.
You should probably only use the classic block as a last resort. Instead, try to learn how to use the other Gutenberg blocks. In the future, the classic block could be deprecated.
Convert to Blocks
When you're ready to use the new Gutenberg blocks, you can convert your classic text into blocks.
Use the three-dot menu to the right of the classic block, and choose "convert to blocks."
WordPress will automatically figure out what type of content is within your classic block, and convert it to the appropriate Gutenberg block (paragraph, list, table, heading, etc.)
Edit as HTML
Another thing you can do with the classic block is edit your content in HTML. Use the three-dot menu and choose "Edit as HTML."
You'll now see all the HTML code that makes up your content, and you can edit it freely. This is the same as using the "Text" tab (as opposed to the "Visual" tab) in the classic WordPress editor.