I was just going to post something along similar lines. I live in Sweden and don't know how to help usefully, but if anyone ends up hereabouts and needs a place to stay, let me know at johan.domeij@gmail.com
Sweden does not have a sponsorship program like you describe, but seems positive to taking refugees (in Swedish, sorry. Suggestive, but in no way firm confirmation. I strongly believe refugees from Ukraine would be let into Sweden at this time, though I have no special knowledge beyond being a Swedish citizen.)
In Slovakia, Ukrainian citizens without visa can stay for 90 days (without visa = no work, no health insurance). If you come with kids, you need some certificate that you are their relative.
You can ask for asylum here, but if you do, then you are not allowed to go to other EU countries until you get an answer, so maybe think twice and ask for advice before you do.
More info in Ukrainian language here.
Here is an update on our efforts in Canada.
1. There are nearly five of us who would be willing to sponsor a refugee to settle in Canada (indefinitely or for however long the war might last). There is a requisite amount of money that must be committed beforehand to cover at least a few months worth of costs for settling in Canada and living here for a few months. Determining whether 3 or more of us would be able to cover those costs appears to be the most significant remaining bottleneck before we decide whether to take this on.
2. There are two effective altruists in the province of Alberta who would be willing to sponsor another refugee if another may need that help. If you or someone you know is in touch with someone living in Alberta, in particular the Calgary or Edmonton areas, who might be willing to sponsor as a refugee a community member for Ukraine, please reply with a comment or contact me.
Some rationalists and effective altruists in British Columbia have begun mulling over the option of whether they would be able to sponsor a fellow community member living in Ukraine to stay as a refugee in Canada. There is a process by which 5 or more citizens of Canada can sponsor a refugee. Have you or anyone else you know in the community begun considered doing something similar or other ways to aid peers still in Ukraine?