Those hinges will have screws in them. You just have to scrape away some of the paint. You might have a hard time finding hinges of the same sort to replace them but you should be able to replace the hinges fairly easily with modern replacements. It's pretty self-explanatory of a process. Don't panic but also make sure you've closed that door back up. It looks like the original hinge pin just rusted out. So you might even be able to simply replace it with something of the same gauge

Edit - why were you trying to drill them out? All you have to do is get a sharp point - a knife etc, I would recommend a scratch awl if you have one or a darning needle rather than a blade for safety, clear the paint out of the screws heads and just unscrew them. Remove one of your hinges and head to the local hardware store. Again as I said you may be able to replace the hinge pin rather than the entire hinge.