Yes, Obsidian by itself reads and creates files in a folder of your choice. The folder is considered a "workspace" and you can create/switch to other workspaces if you want. There's desktop apps and mobile apps. Aside from Markdown files, you can store and display images, audio files and video files (haven't tried that last one though).

However, you can choose to backup and sync your workspace across multiple devices using a cloud service. The most pain-free way to do this is by buying a subscription to Obsidian Vault. This is also a good way to support the development team, since the main product is free. When you store your workspace on Obsidian Vault, you can choose to specify an encryption key, so that theoretically nobody at Obsidian should be able to read your data.

But there's nothing stopping you from using other cloud storage services. There's even a community plugin you can find to use a Git repo for storage.

They have a paid cloud service but by default it just stores markdown files locally. Unlike many apps with that model, they're not at all pushy about their paid services in the app.