Emirates will only take GCC (Gulf Cooperation Council) nationals. In otherwords if you come from one of the Gulf states then you would be able to apply. Emirates has a large proportion of ex-pat pilots from round the world so their sponsorship scheme is their way of getting native flightdeck crew to try and even the balance. Gulf Air are in exactly the same situation and possibly Qatar Airways as well. Etihad from Abu Dhabi will no doubt follow suit with a similar sponsorship scheme as well as they continue to expand.
However Emirates is one of the very few airlines that is very profitable as well as having a very full order book. Their need for experienced pilots will continue for many years. You will normally need medium-heavy jet experience to get into Emirates with a few thousand of hours. It has to be one of the if not the best company out there at the moment. There is no harm in having a goal to aspire to!