Solutions do exist locally. Globally they MAY blow up as you already know.
The blowup will be dominated by the deterministic system. You did not write what is your initial condition - note that $0$ is a perfectly fine solution to your equation. If your initial condition is say $X_0=1$, then it may blow up, but with positive probability the solution may actually converge to $0$. This is because if the $X^2$ drift term were not there then the solution would be $X_0 e^{B_t-t/2}$ which converges to $0$, and if you add your drift, as long as $X_t$ is small then the drift is smaller than $+\delta X_t$ (e.g., as long as $X_t<\delta$). But the solution of the equation
$dX_t=\delta X_t dt+X_t dB_t$ converges to $0$ as long as $\delta$ is small enough
(in fact $\delta<1/2$ will work).

So with positive probability, starting from any $X_0$ you become smaller than say
$1/4$ after a finite time, and then with positive probability you actually converge to $0$ (using comparison theorems for 1D SDE's as in Ikeda-Watanabe should be enough to prove this). On the other hand, with positive probability you blow up. So I am not sure what do you mean by ``moments of the blow up time''.