OK - turns out I was wrong.

There is a 24 hour transit visa available which allows you to leave the airport - it has to be applied for at a Qatari embassy (unless provided as part of QR's STPC package, which obviously does not apply in this case)

I won't provide a link as they generate massive URLs which probably are valid only for one visit each: visit http://portal.www.gov.qa, type "Transit Visa" in the search box, and click on the first search hit.

No idea how much this visa costs - but in terms of time and hassle, it's probably just easier to get the Visa on Arrival (which allows a stay of something like 28 days).