In almost every case, yes, you can exercise (purchase) your stock options.
But, why would you?
At this stage, you cannot sell them (generally speaking sales are highly restricted pre-IPO, and they're not worth anything yet anyway), and there is always the probability that some other thing will happen to negatively affect their long-term value.
Personally, I've never seen the value in exercising stock options while you're still in the privately-held stage of the company. There is far more downside than upside in most cases. In certain scenarios, you can avoid short-term gains taxes by exercising then waiting 1+ years before selling them. This is still rarely worth the risk overall unless you are positive of the outcome, and it's going to save you a significant chunk of money (>$100K, IMO).
My unsolicited non-valued advice: Let your options continue to exist on paper. If you are contemplating leaving the company, it might make sense to explore exercising them. But then again, why would you leave if the stock was going to continue to increase in value? ;)