Yeah, absolutely. There are a bunch of places. Every big moto sales/repair shop will offer a moto hotel of some kind, because a lot of people live in apartments and don't have their own garages.
For example, this one is very close to Tallinn Airport: BIKEPoint - Talvehoid and charges 1 euro per day for a regular-sized bike, 50 euros minimum charge. They are also a retailer of used bikes (btw non-residents can buy and register vehicles in Estonia), a repair shop, and a Kappa luggage dealer - I bought my current bike from them and can say they are a decent company.
Some other links available at MOTOHOTELL. Or just google "motohotell tallinn" and use Google Translate.
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