Walking from the harbour up towards the other restaurant front along Elizabeth Street, you'll come across Mako Fresh Fish. Apparently it's quite well known for its fresh seafood and the Chinese tourists seem to like it. Not sure about the second part, but the first seems to be true.
There are two counters within the same store - a fresh seafood spot and a cooking area where fish and chips etc. can be purchased. Given it was around 4pm and I just needed something to tide me over until dinner, it was mainly a snack I was after. Of the fresh counter, the oysters seemed best.
When reading menus online, Hobart restaurants charge about $35-40 for a dozen oysters, even natural. By contrast, the same ones are around $14 at Mako. I sat down, paid $1 for a lemon (which was unnecessary in hindsight) and enjoyed the large, creamy, moderate strength delicacies at the central table. I also bought an arancini ball with cooked salmon inside. They were kind enough to microwave it for me but unfortunately the outside was hot but inside still cold. I won't make that mistake again.
It'd be a great spot for a full lunch (or dinner) - oysters from the counter, fish and chips (which is a bit pricey) from the cooked area.