Mount Field National Park
Mount Field is a bit more than an hour drive from Hobart. It has plenty of shaded peasant easy walks. Even if you are not very fit, try the Russell Falls walk. It is a minimum effort, maximum reward kind of walk. You need to pay for an entry unless you hold an annual pass. There are some very rustic cabins you could book on their website for a fee.
http://www.parks.tas.gov.au/?base=3589
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