The short answer is no. At least at this time.
Refugees are people with a well-founded fear of persecution because of race, religion, nationality, political opinion or membership in a particular social group. This definition comes from the Refugee Convention, which most of the world's countries have ratified. Refugees receive the designation of refugee (often with the assistance of UNHCR or a partner organization), outside of their home country and outside of their country of final destination. Fewer than 1% of the total world's refugee population is even considered for possible resettlement in a 3rd country, and the process is very long, often several years.
Asylum seekers are people who are either at the border or already in the country and they request that they be permitted to seek asylum. Asylum seekers must also ultimately prove a well-founded fear of persecution on account of one of more bases - race, religion, nationality, political opinion or membership in a particular social group.
A person seeking asylum bears the burden of proving that:
1) they have a well-founded fear
2) of being persecuted (had/have credible threats of state-sanctioned harm or actual harm)
3) for reasons of race, religion, nationality, membership in a particular social group or political opinion
4) the government is doing the persecution, or is ignoring or acquiescing to or unable or unwilling to help prevent a non-governmental group from persecuting you.
War or civil strife is not a basis for asylum or refugee status. War or civil unrest may be so dangerous to civilians that it can give rise to a right to temporary protection status (TPS) or other humanitarian status, if provided by the government.
Right now, the U.S. administration is working on a plan which would allow vulnerable Ukrainians, like activists, journalists and those who are part of the LGBTQ community, to safely enter the U.S. at least temporarily. It would also speed up the reunification of Ukrainians with U.S.-based family members. This should be clarified within the next few days.
If you are at the border, inform the border officers if you fall into the categories below, as you may qualify for special protection:
• A minor traveling alone
• Over 65 years old
• Alone with your children
• Suffering from a serious illness or disability
• Pregnant, or have had a baby in the past 3 months
• Victim of torture, or physical, psychological or sexual violence
• Injured or disabled
• Victim of trafficking (labor exploitation, including sexual servitude)
• You have a family member who is legally in the country where you are seeking asylum [note; family reunification processes run parallel to asylum processes, and also take time]
You should be ready to answer these questions at the border:
• What happened to make you leave your country (persecution or fears of persecution for reasons of race, religion, nationality, politics or social group)
• Did other people like you have had similar problems?
• Why did you have to leave your country instead of relocating within it and why you can't go back?
It is advisable to travel with the following documents and anything documenting danger and/or persecution: passport, ID, medical records, letters, police reports or threats made in writing. It may be safer to not carry anything except passports, and instead, email yourself and a trusted individual copies of these documents.