Stork House was the Stork Hotel back when the Bristol Port Railway and Pier, which connected Hotwells with a deep-water pier at Avonmouth, was in its heyday.

The Stork Family Hotel at Hotwells sought custom by advertising to seamen on ships arriving at Avonmouth, a combined rail ticket to Bristol and meal in this hotel. The same concern also advertised bed, breakfast, hot evening meal and seven days’ ticket to Avonmouth by BPR for sixteen shillings a week.

— Colin Maggs, The Bristol Port Railway & Pier, Oakwood Press, 1975

I imagine this would have been quite an attractive deal to a sailor, who could stay in Hotwells with its vast plethora of pubs and other entertainments and pop back to his ship in the comparative wastelands of Avonmouth when necessary. (There would also have been plenty of sailors who were already quite used to hanging around in Hotwells while their ship was in town rather than waiting around at the new pier at Avonmouth, and familiar with the local facilities. Thinking about what other "facilities" there may have been for sailors makes me wonder again if the bit of the Hotwell Road that ended up being called "Love Street" might've been a bit of euphemism at some stage or other...)