I think you can study and work. I think this way because to stay in Ukraine either for studies or work it requires to obtain a "D" type visa and then a temporary residency certificate.
But in case of employment you will need to obtain also a work permit. The last document is not merely a formality. Your employer will need to prove that there are not local fellows able to perform a job, plus to show that your qualification corresponds to the job requirements. So if you want to do jobbing here or some other work that does not require a degree, then it will be very hard for you, imho, to get officially employed. However, it is Ukraine, and unofficial jobbing is widespread here. You just should be cautious and try not to get in troubles.
