The big one is Øvre Pasvik National Park, about 100 km south along the Russian and Finnish borders — a last fragment of Siberian taiga inside Norway, with brown bears, elk, pine forests and the Treriksrøysa cairn where three countries meet. Well-marked hiking trails and cabins for overnight stays. In winter, the same country is crossed by snowmobile trails.
The Storskog border crossing at 15 km east is worth a look even if you cannot cross: the infrastructure of a Cold War frontier, now handling (or, since 2022, mostly not handling) Russian-Norwegian transit traffic. The coastal village of Jakobsnes, on the far side of the fjord, is reachable by a short local ferry in summer and makes a quiet afternoon away from town.