In the 30's, "wild tourism" of many types became prohibited, but it became legal again in the 60's following the Khrushchev Thaw. Hiking, camping, kayaking, mountaineering, and other outdoor activities became very common in the USSR between the 60's and 80's. In 1966, the literary gazette announced that the word "savage" could be used without quotations to refer to wild tourism. In the early 1970's, a survey determined that only 25% of vacationers had permits. In the 60's and 80's, tens of millions of people went on officially unorganized travels, in what was called a recreational boom.

There were probably some places where you'd require a permit, but for the most part you could go wherever you wanted. Hikers also made their own equipment (my dad used to make crampons for climbing on ice), and there was even a musical genre associated with this type of outdoor activity.

Sources:

We Seek That Which Wasn't Lost: The Soviet Savages seeking a place under the sun — a well-researched article about the wild tourism boom (in Russian)

Activity Outside The State: Self-Organization of journey tourists — an article that deals with the more lengthy trips like the one you're describing (in Russian).