Recall the definition of standard state

The standard state of a substance means its most stable form at a given temperature and at a pressure of 1 bar.

Important point: standard state does not fix the temperature to $273\,\text{K}$ or $0^\circ C$. The temperature must always be specified separately, commonly $298\,\text{K}$.

Check option A

Option A says:

The standard state of a pure gas is the pure gas at a pressure of 1 bar and temperature 273 K.

This is incorrect because standard state requires pressure $=1$ bar, but temperature is not necessarily $273\,\text{K}$.

So, A is false.

Check option B

Option B says:

$\Delta_f H^{\circ}_{500}$ is zero for $O_2(g)$.

The standard enthalpy of formation of an element in its most stable standard state is taken as zero at that temperature.

At $500\,\text{K}$, oxygen in its standard state is still $O_2(g)$.

Therefore, $\Delta_f H^{\circ}_{500}[O_2(g)] = 0$

So, B is true.

Check option C

Option C says:

$\Delta_f H^{\circ}_{298}$ is zero for $O(g)$.

This is incorrect because the standard state of oxygen at $298\,\text{K}$ is $O_2(g)$, not atomic oxygen $O(g)$.

Since $O(g)$ is not the standard state, its enthalpy of formation is not zero.

So, C is false.

Check option D

Option D says:

The term 'standard state' implies that the temperature is $0^\circ C$.

This is incorrect. Standard state does not imply any fixed temperature like $0^\circ C$.

So, D is false.

Conclusion

Only option B is correct.

$\boxed{B}$