Can you still play World of Warcraft without the Burning Crusade expansion pack?

It's not a problem. There's really no reason before level 58 that you'd want the expansion anyway. Yes, there is a new race for each faction and a new profession but these are not necessary and just to get a feel for the game with your first character you'll end up knowing what you REALLY want to be by the time you are able to pickup the expansion when your budget allows for it. You'll need a bit of time to go from level 1 to 58 anyway. Save up and you should have the extra money in no time.

Happy Hunting!