Definitely possible as there are so many hiking options in TAS. Check out the Parks and Wildlife Service list of 60 'short' (up to day) walks:
https://www.parks.tas.gov.au/index.aspx?base=1315. I would add Mt Anne as a BIG day walk, or Cathedral Rock and Collins Cap as other great options in the vicinity of Hobart. So you can easily find day walks with B&B or 'shack' (cabin) accommodation nearby, and also many private campgrounds include cabin accommodation. A lot of owners use airbnb to advertise their second homes too. Having said that, some parts of TAS feel very remote and have limited accommodation to retain their character, so for example around Cradle Mountain you either need to book well in advance and accept the elevated cost, or accept a drive in and out for ~1 hour each day. I do recommend that you find accommodation in more rural areas or smaller townships for much of your trip. The wildlife in TAS is fabulous but mostly nocturnal. You don't really want to be driving after dusk, but if you can stay in a rural area you will see more.
I would recommend you aim for 10 days and a circuit to see the east, north and west coasts and a few days south of Hobart, as they are all varied. If you have 7 or less days, just do one way Hobart-Launceston (or visa versa) and choose east or west. Depending where else you are doing on the mainland does dictate things you should try to see while there. For example, in the central northern part of TAS, there is good food and wineries, plus a good chance to see platypus in the Tamar area and mobs of kangaroos at Narawntapu NP. But you may be seeing 'similar' things outside of Melbourne or Sydney, depending on your plans.