Variance of White Gaussian Noise It could seem an easy question and without any doubts it is but I'm trying to calculate the variance of white Gaussian noise without any result.

The power spectral density (PSD) of additive white Gaussian noise (AWGN) is $\frac{N_0}{2}$ while the autocorrelation is $\frac{N_0}{2}\delta(\tau)$, so variance is infinite?
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.