One of the closest hiking options in Ushuaia is to do the relatively easy to moderate trek up Martial Glacier. The initial ascent can be done the short and steeper way, or the slightly longer and easier way. I choose the steeper way and it really isn't terrible. At the top you're rewarded with a terrific view of the city below, which is quite tiny.
Named in honor of explorer Luis Fernando Martial, who led a French scientific expedition in 1883, the glacier is located pretty close to the city. It's easy and affordable to get a cab ride there, and when you're done, one of the coffee shops at the base can easily radio a taxi to come pick you up.
In the winter, there's a small ski slope here complete with a chair lift, but in the summer the lift is not operational. With the path clear of snow and ice in summer, there's not much to worry about except for how long you should stay and explore the area.
The path leading to the top of Martial Glacier.