Although it is impossible exaclty, there are approximate constructions, based on approximate values of de π.
The most famous approximatie value is π ≈ 22/7 = 3.1428... instead of 3.141592...
A possible construction with that value might be :
EB = R/4, AF = AB, OP parallel to EF, AC = AP.
AE = R + 3R/4 = 7R/4
AC = AP = AO × AF/AE = R × 8/7
Then BC = R × (2 + 8/7) = R × 22/7.
Circle with diameter BC and perpendicular in O to BC intersect in D. BD is the side of the equivallent square : BD² = BO×BC = R² × 22/7
Relative inaccuracy is (22/7 - π)/π ≈ 0.0004 that is less than 1 in thousand (practical construction errors are even worse !).