Variance of White Gaussian Noise
White Gaussian noise in the continuous-time case is not what is called a second-order process (meaning $E[X^2(t)]$ is finite) and so, yes, the variance is infinite. Fortunately, we can never observe a white noise process (whether 
Gaussian or not) in nature; it is only observable through some kind of device, 
e.g. a (BIBO-stable) linear filter with transfer function $H(f)$ in which case 
what you get is a stationary Gaussian process with power spectral density $\frac{N_0}{2}|H(f)|^2$ and finite variance
$$\sigma^2  = \int_{-\infty}^\infty \frac{N_0}{2}|H(f)|^2\,\mathrm df.$$

More than what you probably want to know about white Gaussian noise
can be found in the Appendix of this lecture 
note
of mine.