It is, or will be in the near future, a common problem in most countries that visiting foreigners are virtually unable to leave even though their visa or right of residence expires. Some countries have already reacted and officially suspended visa requirements. For example in Germany, visa requirements for foreigners already present in Germany have been suspended for most practical reasons until June 30.
Other countries are of course also aware of the practical problems, but the current situation seems so exceptional and unexpected that it was not taken into account when drafting laws on immigration. Depending on the legal system, it can take a long time to change a law and formally find a solution to current problems. I cannot find any statement from the Ukrainian authorities that the visa requirements have been suspended there, but in their Corona FAQs, the Ukrainian Foreign Ministry at least states that it is aware of the problem and is looking to find a solution:
Since the issue of expired documents can be a common problem during the quarantine period, the MFA has already contacted foreign governments to resolve the issue so that all concerned are not subject to sanctions or other measures punitive as a result of such expiration. The MFA will provide additional information as soon as the necessary arrangements are in place.
The declaration is taken from a section on the expiration of work permits, but with goodwill it can be read as applicable to all kinds of travel documents and permits. I would usually have recommended that you contact the local immigration police (or other immigration authorities) to find out what to do, but with my experience with the Ukrainian authorities, I doubt you will get a reply.
I am not up to date on the details of travel restrictions in Ukraine, but if it is illegal to travel between provinces at the time of writing, this will of course also apply to cyclists. Police checkpoints are even quite common in "regular period" in Ukraine and I would not expect them to be less frequent now. It is unlikely that you can cycle 500 km across Ukraine without attracting the attention of the police. However, I find current information from authorities of other countries that it is assumed possible to leave Ukraine by land, so there could be exceptions to restrictions prohibiting travel between provinces, for example for foreigners who return home. Unless you are a very keen cyclist, you will need several days for the trip and with restaurants (and hotels?) Closed, this does not seem to me to be a very good idea.
The most obvious thing to do, as the comments also suggest, is to contact the embassy of your home country in Ukraine and ask if they can suggest a way to get home. . I guess you've tried this before?