A game could be as big as 40gb so you might run out of space quick. Use SSD for system an a 1tb hdd for games. They go for about $50. For a budget build you might want to leave the SSD out if it stops you from getting extra space.
A 240GB SSD is enough for the base system and 2-4 big titles. If you are willing to trim your file storage down, it's an acceptable drive. But you will not be comfortable with it for long term. Add a 1TB HDD or more SSD space later down the line, but you can probably make do with only 240GB for a year.
I run windows and my games off of my SSD, then I move the games to HDD storage when I take a break from them. If you are willing to re download games, it's not too bad to just have an SSD.