I travel the Exeter/Torbay area on business, and there are few round the Exeter/Dawlish area. This might seem a long way from you, but in reality it's only a quick trip up the A38 from you, and would have the advantage of being quite close to both the A38 and M5 for a start point for your touring. Dawlish is only 18 minutes from Exeter after all.
Try Lady's Mile Caravan Park Dawlish. Tel 01626 863411, for starters. I was talking to the guy there only a month ago about this, and he told me that they do storage, but if he can't help, their reception know of several farmers who have storage close by. Well worth a chat with the girls/guys on the reception.
There is actually a storage site on the other side of the road from Lady's Mile park. I don't know if they have space, but it's nothing to do with this caravan site. They may well know the number.
Good luck. I know space is tight in the South Hams. If the worst comes to the worst, a lot of sites might offer you temporary winter storage during the quiet season, which would at least give you a bit of breathing space to look around.
I'll keep my eye open for you today as I do my rounds, and see if I can spot any others.