It is, or will be in near future, a common problem in most countries that visiting foreigners are practically unable to leave although their visa or right to stay expires. Some countries have already reacted and officially suspended visa requirements. For example in Germany, visa requirements for foreigners already present in Germany, have for most practical purposes been suspended until June 30th.
Other countries are of course also aware of the practical problems, but the current situation seem to be so exceptional and unexpected, that it was not accounted for when immigration laws were written. Depending on the legal system, it may be a time consuming process to change a law and formally find a solution for the current problems. I can't find any statements from Ukrainian authorities that visa requirements have been suspended there, but in their Corona FAQs, the Ukrainian Ministry of Foreign Affairs at least states that they are aware of the problem and are seeking to find a solution:
As the issue of expired documents may be a common problem during the quarantine period, MFA has already contacted foreign governments to settle the issue so that all people concerned do not face penalties or other punitive measures as a result of such expiration. MFA will provide additional information as soon as necessary arrangements are in place.
The statement is quoted from a section about expiring work permits, but with some goodwill, it can be read as applicable to all kinds of travel documents and permits. I would usually have recommended you to contact the local immigration police (or other immigration authorities) to ask what to do, but with my experience with Ukrainian authorities, I doubt that you will get an answer.
I am not up to date on the details of the travel restrictions within Ukraine, but if it is illegal to travel between provinces as you write, this will of course also apply to bicyclists. Police checkpoints are even during 'regular times' quite common in the Ukraine and I would not expect them to be less frequent now. It is quite unlikely that you will be able to bicycle 500km through Ukraine right now without attracting any attention from the police. I do however find current information from other countries' authorities that it is supposedly possible to leave Ukraine by land, so there might be exceptions to the 'no travel between provinces' restrictions, e.g. for foreigners heading for home. Unless you are a very avid bicyclist, you will need several days for the trip and with restaurants (and hotels?) closed, it does not sound like a very good idea to me.
The most obvious thing to do, as also suggested in the comments, is to contact your home country's embassy in Ukraine and ask if they can suggest a way for you to return home. I suppose you have already tried that?