Kaunas train station:
Self service storage terminals. This service is available 24/7; however, you can only place your luggage for storage or collect it from lockers within working hours of the Railway Station.
Railway station opening hours: I–VII, 04:10–21:45
Website: Luggage storage at train stations | LTG Link

Kaunas Bus Station:

The storage service is provided by the delivery/shipment company – "Siuntos autobusais".

Website: Kaunas bus station | Kautra
Email: info@siuntosautobusais.lt
Phone number: +370 37 409 069
Working hours: I–VI 05:45–19:30; VII 07:15–18:30