I know it's possible to export to 256 colours and to change colour mode to 256 colours, flattening the image. But I'd like to be able to setup the image to work only in 256 like you can set it up to work in RGB or HEX. Pretty sure this used to be possible, but can't figure out how to do it in PS5.
To convert a document to Indexed color mode, use
Image > Mode > Indexed Color. You can select from a variety of choices (web, Windows OS, Mac OS, etc.) as a starting point. As Horatio points out, this immediately disables about 95% of Photoshop's capabilities: you forfeit layers, filters, channels, adjustment layers and masks, and you have only a limited selection of destructive adjustments available.
If you don't mind working within these limitations, fair enough, but I would strongly recommend a different, non-destructive and far more flexible approach.
Work with your document in RGB mode, but keep a 256-color Posterize adjustment layer at the top of the stack. This leaves all of the regular tools of Photoshop at your disposal, but still gives you an immediate and accurate view of your final output.
When you're ready, save your PSD then convert to Indexed Color and save separately.