Upvote:5
Assuming you are an Indian National, you are not on the list of countries with Visa on Arrival. The same site says that you can arrange a 96-hour transit visa with the airline, which I suspect means that you do not have to stay in the airport. I suggest contacting FlyDubai to find out how to get this visa.
There are special rules in the Gulf Countries about unaccompanied young women from Russia and other former USSR countries, but as far as I know, they do not apply to India, nor to Romania (Romanians are entitled to visa on arrival).