Step 1: Recall what "eutectoid" means geometrically.
On the iron-iron carbide diagram, the eutectoid point is simply the single carbon composition at which the horizontal line at $723^\circ C$ touches the boundary of the austenite field exactly at its lowest point, the bottom of the "loop".
Step 2: Why this composition is special.
At any other carbon percentage, cooling austenite through $723^\circ C$ first passes through a two phase field (austenite plus ferrite, or austenite plus cementite) before the eutectoid reaction can happen, so some proeutectoid phase separates out first. Only at the exact eutectoid composition does 100% of the austenite transform directly and completely into pearlite with no proeutectoid phase at all.
Step 3: Reading off the value.
That special composition on the standard $Fe-Fe_3C$ diagram is read as approximately 0.76 to 0.8 percent carbon by weight, universally rounded to 0.8% C in textbooks and problems. The other values given, 0.008%, 0.18%, and 2.1%, correspond respectively to the solubility limit of carbon in ferrite, a typical mild steel, and the steel to cast iron boundary, none of which is the eutectoid point.
\[ \boxed{0.8\% \ C} \]