Is there a baggage room at Kiev main station? If not, where can I store my luggage for some days?

There is a baggage room inside the station. It is down a passage way. You buy a ticket from one side of the passage for the baggage you want storing. You then cross the passage with your ticket and baggage and store it.
The baggage facility is run by people and not machines and lockers.

There are three luggage storing facilities at Kyiv-Pas., two automatic lockers (in the basements of east and west wings of the station) and one 'human operated' storage room (also in the basement, in between wings, but closer to the east one). As far as I remember in automatic lockers you can store things for up to 48 hours, in 'human operated' maybe more, not sure.
Be sure to check the times of the human operated baggage storage as they are often closed for an hour around lunch and dinner.
Ruth
As far as I remember in automatic lockers you can store things for up to 48 hours, in 'human operated' maybe more, not sure.
In the human operated storage you can left your baggage for any time, just warn a clerk in advance.
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Oksana @ActiveUkraine

My partner and I arrived at the Kiev main train station early in the morning with 2 large backpacks. We saw someone sweeping and mopping the floor nearby so we asked her where to store our backpacks. This uniformed Ukrainian girl took us to some kind of employees storage or utility room and we paid her 5 hryvni for each backpack. We were not worried because there were only old clothes in our backpacks. We went back to the train station around 5 p.m. and retrieved our backpacks, no problem.

I recently used the automatic lockers in the east wing to dump my backpack whilst I waited for an overnight train; buy a token (for 15 hryvni) from the counter

except for the clerk having a break (always ask them when they have their breaks if there is not already a sign on the wall), that you might experience to be charged for 2 days if you leave your backpack from 11.30 PM to 0.30 AM (price x calendar days), that they don't understand your name and write "Estonian" or something like that instead;) or that you have lost your token and have a hard time convincing them that it is your backpack you are very unlikely to have any problems. The people operating these places are normally as serious about their jobs as your granny with her baking if you understand what I mean:)
Good luck
J