Unfortunately, fleeing war is not a sufficient qualification for refugee status in the United States. It will be very difficult for individuals who cannot demonstrate past persecution or a well-founded fear of future persecution in Ukraine because of their race, religion, nationality, membership in a particular social group, or political opinion to qualify as a refugee under U.S. immigration law.
But if you believe you have a legal basis to file for asylum, it will be best to speak with an immigration attorney about your specific situation. Currently, the process of applying for asylum can take 3-4 years or more.