The solution to your problem is data encryption. On Linux, there are different levels of data encryption, whether you want to encrypt the whole hard drive or only the /home partition (where all of your private files will be stored).
The ArchLinux Wiki has a very detailed page [1] about all the available options for disk encryption.
Regarding Ubuntu, when you install it on your system, it will ask if you want to encrypt your /home partition [2] or your whole disk [3].
I got my laptop stolen in France last summer, and as much as this was a pain in the ass, at least I didn't have to wonder if the thief had access to my private data (photos, documents and the like) since the /home partition was encrypted. Hell, the thief probably had a WTF moment when (1) he discovered it was not a French version of Windows but an English version of Ubuntu running and (2) when he discovered it was a Taiwanese laptop with this keyboard layout [4]. Good luck to sell that on the French black market :)
[1] https://wiki.archlinux.org/index.php/Disk_encryption
[2] https://www.howtogeek.com/wp-content/uploads/2012/06/ximage8...
[3] http://www.tecmint.com/wp-content/uploads/2016/02/Ubuntu-16....
[4] https://c1.staticflickr.com/8/7501/16104079539_00c39c200d_b....