Comment by Gro-Tsen on What's the earliest result (outside of logic) that cannot be proven constructively?

The very first postulate in Euclid's Elements is “to draw a straight line from any point to any point” (“ᾘτήσθωἀπὸπαντὸςσημείουἐπὶπᾶνσημεῖονεὐθεῖανγραμμὴνἀγαγεῖν”). The fact that any two points in the plane $\mathbb{R}^2$ are connected by a line is equivalent to the robust division principle, which is not provable constructively.