Not including the game, it's quite possible. I've been building my own budget builds for years and I'm sure we've all seen it done; but for BF4, you ARE asking too much.
Assuming, you can reuse a case and it's decent PSU and it's decent mobo, we're talking: $80 Quad-Code Athlon II CPU, $80 2x4GB RAM, $80 120GB SSD, and at least whatever decent $80 graphics card you can get. Don't forget the game: $40 on sale.
Maybe you can hit MID @ 720P? $360 Total.
You haven't told us anything you might have lying around you might be able to use. Better yet, if you like @Radeon's build, then choose what your willing to skimp on, and downgrade those parts as you fit best. Though he has a good selection of what your bare minimum should look like.