Can the GARCH intercept be 0? Reading the ARCH and GARCH theory I understood that 
alpha_0 have to be > 0 , but when I estimate my GARCH-X(1,1) model I obtain a non significant constant, like this:

Is it a problem? How can I justify that? Is it "against" the ARCH theory?

It makes sense to have a nonzero constant in the conditional variance equation unless your restrict $\alpha_1+\beta_1=1$ to obtain the exponentially weighted moving average (EWMA) model. (For a brief introduction to EWMA, see Alexios Ghalanos' very short blog post here). Otherwise, your fitted conditional variances will be decreasing towards zero over time, and you probably do not want that.
Obtaining correct standard errors for the coefficients of the conditional variance equation is also nontrivial; the software implementation may or may not be doing this correctly. See Francq & Zakoian "Testing the Nullity of GARCH Coefficients: Correction of the Standard Tests and Relative Efficiency Comparisons" (2009) or chapter 8 of Francq & Zakoian "GARCH Models" (2nd edition, 2019). (Unfortunately, I think the focus there is on the $\alpha_1$ and $\beta_1$, not $\alpha_0$... But you may still check them out just in case.)
