N 47.89738 W 90.56047
Two routes to get to Eagle Mountain. Be sure to fill out a day use permit at Entry Point #79.
First route, the Superior National Forest version. From MN 61, turn north on Cook County (Cty) Road 4 (Caribou Trail, MP 92.5) just north of Lutsen. Follow Cty 4 until it ends at FR 170 (17 miles). Turn right on FR 170 and proceed 3 ½ miles to the Eagle Mountain Trailhead. The parking lot is on the left (north) side of road. A total distance 20.5 miles.
Second route is how I get there. Turn on MN 61 at MP 110.3 onto Cook Cty Rd # 12 (Gunflint Trail, on the north side of Grand Marais) and follow it until you come to Cook Cty # 8 (Devils Track Road) (Golf course in on the right, 4.2 miles from Hwy 61). Stay on Cty Rd # 8 until you get to Cook Cty # 27 (Ball Club Rd, 5.6 miles), turn on Cty Rd #27 and stay on it until you get to Forest Road #170 (aka; The Grade, FR #323 goes north here and dead ends), follow FR #170 for 4.9 miles until you come to the Eagle Mountain Trailhead sign. A total distance of 19.6 miles.
From here hike the 3 ½ mile trail to the top of Eagle Mountain. This is the highest point in Minnesota at an elevation of 2,301 feet. Now head to Grand Marais and touch the waters of Lake Superior, but you probably have already done that. If you did, you were at the lowest point in Minnesota at an elevation of around 602 feet. All of this within 24 miles distance.