El Chalten is located about 3 hours drive north of El Calafate. The town is especially famous for the nearby Mount Fitz Roy. El Chalten is the trekking capital of Argentina. So if you want to go hiking, you are in the right place.
It is also used as an alternative to hiking in Torre del Paine, where the campgrounds are often fully booked and hotels are hardly affordable. El Chalten is also rather expensive for Argentina, but in comparison to the Chilean Torre del Paine it is still cheap. But even if the hotels are a little more expensive, the time in El Chalten is otherwise cheap. You don’t have to pay a park entrance fee or something similar. So if you hike the whole day, you have only costs for hotel and food and nothing more.
There are different trails which are possible as day hikes from El Chalten. The 3 main trails, Fitz Roy (10km one-way), Laguna Torre (9+2km one-way) and Loma del Pliegue Tumbado (10km one-way). 2 small trails, Los Condores + Aquilas (3km) and Chorrillo Waterfall (3km). I have described the trails on the respective detail pages more exactly and with many pictures. There is also the unmarked trail to the Laguna Toro (16km), but you have to register at the visitor center of the national parks. I didn’t walk this path myself, so I only mentioned it, but no description.
The water in the park is drinkable, so you don’t have to take too much water with you, because you can refill it on the way. I recommend at least 3 full days to be able to do all the long tours.