I've just joined the Imperial Legion for the first time and this is also my first time doing the Civil War quest. Is there a place where I can sleep and store my things in Solitude?

I need to pull up the Thalmor HQ in the CK as that appears to be a safe place to put stuff. It has a ton of storage. I won't be able to do this until I get home, but you could put something in some of the containers and wait somewhere else for 30 days. Don't wait in Solitude as that may not let the reset timer work as intended. Just to note, it is assigned as someone's home, they just never go there and there is no ownership that I can see on anything.

Just to note, any chests or containers in a "Barracks" are not safe, well the chests are for sure not safe.

Ok I'll check it out but I was really looking for a unowned bed as I'm RP as a soldier and really need to sleep with the grunts.

I think there is a bed down in one of the basemants of the Castle Dour that is unowned and you can sleep on it.