Can Git software (e.g. Gitbox, Github, SourceTree) use a remote repo instead of local?

Question

I like using Git software to push commits, but the ones I use (Gitbox, Github, SourceTree) all ask for a local repo when adding a new repo to them.

Thing is, my repo is

Answer 1

7 years later, the goal is for Git to be able to use a virtual disk, through VFS for Git.

The Virtual Filesystem for Git (formerly GVFS) is an open source system that enables Git to operate at enterprise-scale.
It makes using and managing massive Git repositories possible.

VFS for Git virtualizes the filesystem beneath your Git repository so that Git tools see what appears to be a normal repository when, in fact, the files are not actually present on disk.
VFS for Git only downloads files as they are needed.

This is not (yet) part of Git itself, but:

For that, Git 2.22 (Q2 2019) will help managing such a virtual disk by introducing a new hook "post-index-change", which will be called when the on-disk index file changes: that can help e.g. a virtualized working tree implementation.

See commit 1956ecd (15 Feb 2019) by Ben Peart (benpeart).
^{(Merged by Junio C Hamano -- gitster -- in commit 5795a75, 25 Apr 2019)}

read-cache: add post-index-change hook

Add a post-index-change hook that is invoked after the index is written in do_write_locked_index().

This hook is meant primarily for notification, and cannot affect the outcome of git commands that trigger the index write.

The hook is passed a flag to indicate whether the working directory was updated or not and a flag indicating if a skip-worktree bit could have changed.
These flags enable the hook to optimize its response to the index change notification.