I have solder paste and flux that I regularly use. A lot of times I keep them in my drawer. The solder paste go dry and flux leaks out from the syringe. I have not thought about health risks associated with this practice. Should I store them in a sealed container?
Storing solder paste and flux in a sealed container will help limit possible exposure as well as help prevent them from drying out. I've used something similar to this container, which is relatively cheap and still effective. I also put the solder paste inside several layers of sealed bags before putting it into the container (along with some desiccant packets). All containers leak, some just do so more slowly than others. So there will always be some safety risk.
Storing solder paste/flux at refrigeration temperatures also helps preventing them from drying/leaking. Just don't put any solder paste/flux in a fridge that also contains food or drinks -- OSHA has some strong warning about lead exposure.