I don't know if you need a visa to enter Serbia (depending on which passport you have, what are the agreements between the countries?). However, even if you don't need a visa, I think there is a stay limit: 90 out of 180 days.
There is also an option for a longer stay, which can be obtained through self-employment (opening a company), buying real estate, marriage or for business needs, education, etc.
It is best to ask the Ministry of Foreign Affairs of your country what the conditions are.
Since the beginning of the war in Ukraine, on the streets of Belgrade every day I hear a language that looks like Russian to me (I believe it is not the same as Ukrainian, but I don't know both languages ​​to distinguish them). I don't know if Ukrainians or Russians escaped, but they are younger people, often entire families with small children, who settled here, I believe for longer than 90 days.