I know I can play Minecraft on my raspberry pi, but is Day Z possible or would I need a gaming computer? Though I have a tight wallet ...
We are able to play a version of Minecraft on the Raspberry Pi because the developers of Minecraft ported a version of the game to run on the Pi.
After doing some quick research on DayZ using Google; all indications show that DayZ is a Microsoft PC based game at this time. There was some discussion about Bohemian Interactive supporting game consoles (XBox) in the future.
I recommend that you post your wishes for supporting a version of the game for the Raspberry Pi on the games forum.
If you are interested in using the Raspberry Pi as a gaming computer I would suggest using the Retropie distro.
Although I am unfamiliar with the game DayZ (I would guess zombies ...) you may find games for the retropie which satisfy your gaming needs.