1. **Problem statement:** We have a circle with center $O$ and a tangent line $\overleftrightarrow{AC}$ at point $C$ on the circle. 2. Given that $\angle BAC = 23^\circ$, and since $\overleftrightarrow{AC}$ is tangent at $C$, the radius $OC$ is perpendicular to $AC$. 3. This means $\angle OCB = 90^\circ$ because the radius to the tangent point is perpendicular to the tangent line. 4. Since $\angle BAC = 23^\circ$ and points $A, B, C$ lie on the tangent line, the angle formed between the chord $BC$ and the tangent $AC$ at point $C$ is $23^\circ$. 5. The tangent-chord angle theorem states that this tangent angle equals the measure of the inscribed angle subtending the same chord $BC$ on the circle. 6. Therefore, the central angle $\angle BOC$ subtended by chord $BC$ is twice the inscribed angle $\angle BAC$: $$\angle BOC = 2 \times 23^\circ = 46^\circ.$$ 7. Since the problem asks for the measure of $\angle O = \angle BOC$, the answer is: $$\boxed{46^\circ}.$$