Is it permissible for an employee on a long sick leave to engage in another activity elsewhere, while continuing to receive his full salary from social security?

Working for wages depends on the conditions set by the entity providing the salary (Social Security). If Social Security does not stipulate against working for another entity, then there is no harm. However, if it does stipulate that there should be no other salary, then it is not permissible, due to Allah's saying: "O you who have believed, fulfill [all] contracts," and due to the Prophet's (peace be upon him) saying: "Muslims are bound by their conditions."