Is it possible to build an atmosphere on Mars without an ozone layer? What would trap the gas? Also what is an ozone layer made of?
The ozone layer contains more ozone (O3) than most of the remainder of the atmosphere. It absorbs a lot of UV radiation.
An ozone layer is not required to maintain an atmosphere. All gases in the atmosphere are trapped by gravity only.
If you build a human-livable atmosphere on Mars, the oxygen in that atmosphere will automatically start creating an ozone layer.