As far as I know there’s not any service that lets you stream your files from the cloud without downloading the whole project, as the read/write speeds needed wouldn’t be achievable in a practical sense. What you can do instead is use source control to host your files on the cloud, and then only download the changes you’ve made from other computers on the one you’re currently viewing.
As an example, if you used Github, you could store your project source in a repo (the cloud) and then download that repo to another machine. The first time you go to access it on another machine, you’ll need to download the full project. After that, you’ll only have to download smaller packages that contain only data that pertains to file changes you’ve made, i.e. new textures you’ve imported, level changes, code changes, etc.
I use source control in my project to connect my partner to my code as we live in two separate countries. He typically works while I’m at my day job, then when I get home he’s pushed his changes, which I download to my PC at home, which takes a couple seconds, and then I’m good to go to work, submitting my changes for him to grab the next day, and so on. You can also do this to share your project between multiple computers for yourself, as I also do, taking my laptop with me to work on lunch breaks or other times when I’m away from my desk at home and have some time on my hands.
Let me know if you have any questions.