I created a bare Git repository on a shared drive on my company's server. I have cloned it on my own machine's C drive.
If my machine dies tonight, will everything I have pushed to that bare Git repository be safe? That is, will I be able to just use another machine and clone it, and have everything I've worked on?
In general, the answer is "yes." Any work that you have pushed to the remote will be stored, and can be cloned or pulled into a repository on another system. Any work that hasn't been pushed, whether committed locally or not, will be lost.
As a caveat, how your machine dies can have an impact. As one example, if you have had a failing disk that's been corrupting your data for a while, and have been pushing corrupted data upstream as a result, then the chances are good that there will be corrupted data upstream as well. Because of the way Git stores data, you can usually recover up to the point of corruption as long as you haven't been using forced pushes of rebased branches, but your mileage may vary. Other edge cases like failure during a push are less likely to be problems.
A distributed VCS is resistant to corruption and loss of data, but is not totally immune to it. While it will no doubt be fine in your defined use case, you really ought to incorporate a backup scheme into your high-availability and data integrity planning.