Yes, it can. You may add protocols such as WebDAV to your Nginx or IIS, or FTP to IIS (as far as I know, there is no FTP support for Nginx).
Regarding WebDAV, you can map it then as a drive in Windows on client side.
This being said, of your goal is to be able to store files in the cloud, a better (easier, more reliable) option would be to use an existing service, such as Google Drive.
Since I posted the answer, the OP specified that the purpose is:
To allow specific C# desktop applications to upload user logs to a web location from anywhere around the world.
For logging purposes, one may want to consider solutions such as Logstash, with data stored in Elasticsearch or a similar database, the benefit being that it would be more searchable. Or use a product designed specifically for that, such as Stackify.
Here again, cloud services may be extremely helpful, occasionally providing a C# NuGet package that will make it possible for your application to stream the logs to the cloud provider with as few lines of code as possible. And, of course, you'll get the benefit of being able to search those logs, filter them, or attach specific actions (for instance, sending an SMS when a given message is encountered).
If you want to avoid cloud, still, never ever let users to upload whatever they want to your server. There are, essentially, two reasons.
If the users can upload whatever they want, this may give them a possibility to submit an executable script that may end up being executed on your server, eventually giving root access to it.
With no control as to what exactly is stored on your server, you make it possible to be sued. For instance, hosting viruses, content that infringes copyright law, or child pornography could have very serious consequences in many countries. And “sorry, that's not mine” won't work here, as you are storing this content, and you allowed it to be uploaded on your server.
When you say “Yes it can, you may add...,” may I clarify—this means they can't—until adding FTP, WebDAV etc.?
Those are usually extensions that you have to explicitly add. But they don't require any custom development. You just add them, and configure them accordingly, just as you configure the web server when it comes to serving static files.
But, again, this is a wrong approach in your case. Don't do that.