How short? If I want a really short hoon in the local area I have a "lap" I do when bored and feeling like riding like a loony. If you head down toward Bulford, down the bulford droveway, range road to Tidworth, through Tidworth then take the turning off the 338 for everleigh, before everleigh there is a left turn (just before the old pub) this takes you along the country road to Haxton and Netheravon, as you pass through Netheravon take the turning before the garage that leads up to the army estate, as you pass the kids nursery on the estate turn right, follow that country road and it will lead out back at Bulford droveway (the milston road junction). It's mostly fast straights, with a few decent corners and a couple of blind hills.
Edit - added map