As I understand a Monte Carlo Method, you set up a random experiment so that the probability of a given event is proportional to the quantity you wish to calculate.
For example, to determine the area of a triangle, I would place the triangle in a larger rectangle and my random experiment would be to select a point with uniform random distribution in the rectangle. I would then carry out the experiment a number of times and determine the proportion within the triangle. That number [itex]\hat{p}[/itex] would be approximately the ratio of their areas and so the triangle area would be estimated as the product of the proportion times the rectangular area.
The question of how any you need to obtain a "trustworthy result" is relative to how precise you need to be and how trustworthy you mean. You need to set up a confidence interval for the proportion and map that interval to a confidence interval for the quantity you're seeking to estimate. You can select an a priori degree of confidence (say 95% or 99%) and then decide on the precision you want (width of the confidence interval) and solve for the sample size.
The relation is:
[tex]p = \hat{p} \pm \varepsilon,\quad \varepsilon = z_{\alpha/2} \cdot \sqrt{ \frac{\hat{p}(1-\hat{p})}{n}}[/tex]
where [itex]\varepsilon[/itex] is your margin of error with [itex]\alpha[/itex] being the degree of uncertainty, one minus the confidence. Thus for 95% confidence [itex]\alpha = 0.05[/itex]. The [itex]z_{\alpha/2}[/itex] is a critical z-score, tables of which you can find on the internet.
You can, for purposes of solving for n, work with the "worst case scenario" where the proportion is about 1/2.
[tex]\varepsilon = z_{\alpha/2}\sqrt{\frac{1}{4n}}[/tex]
so
[tex]n \approx \left(\frac{\varepsilon}{4z_{\alpha/2}}\right)^2[/tex]
Note that this [itex]\varepsilon[/itex], margin of error is for the proportion and you must scale it for/from the actual quantity you seek to determine.