I just joined a new team (very small: four developers in total, with two of those leaving soon). The two original developers set up the git repo on a folder in a Windows network share.
Am I taking crazy pills, or is that a bad idea? Our organization does have github/gitlab/bitbucket available, so is there any good reason not to use those hosted solutions?
It is fine. But for your personal growth and some more peace of mind, you should migrate the repository to one of the git services of your organization.
Personal growth because you will be able to use feature like branch policies, CICD pipelines or an integrated work item board.
Peace of mind because that network share is less likely to be recoverable than a self hosted gitlab instance for example.
The migration process also only take less than 5 min so there are more advabtages than disadvantages to do so.