Just realised, they may have meant this globe, which is lovely (but further out from exeter):
http://www.globehotel.com/index.html
I'd stay there and then cycle into exeter from Topsham along the canal, stopping off at Turf locks (or double locks) for some ale/food:
http://www.turfpub.net/index.php http://www.doublelocks.com/
This may not fit in with your route mind, but would be very "quaint".
@JonoMarshall started
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Just realised, they may have meant this globe, which is lovely (but further out from exeter):
http://www.globehotel.com/index.html
I'd stay there and then cycle into exeter from Topsham along the canal, stopping off at Turf locks (or double locks) for some ale/food:
http://www.turfpub.net/index.php
http://www.doublelocks.com/
This may not fit in with your route mind, but would be very "quaint".