Which free/cheap tools can open ESRI Geodatabase Files?

To determine which type the data is provided as just download a copy and look at the extension. If it ends in .mdb it's a personal geodatabase, which is really just a Microsoft Access database. If you want you can open it up and poke around in the tables, you may be able to find what you need there. If it shows up as a folder with .gdb at the end of the name then it's a file geodatabase and you can only access its contents with some GIS specific software.

The good news is that there is free software that can open up either format and let you use it. The learning curve is a bit steep, but that's true whenever you delve into an entirely new area. The software is at: http://www.qgis.org/

I, unfortunately, am not very adept at using QGIS so I can't offer specific instructions on how to open a geodatabase with it. But, I hope that gives you a starting point that will be of some use. This site undoubtedly has some old questions that will get you off the ground. The keywords you will want to search with are 'QGIS', 'Geodatabase'. You can also find tutorials if you poke around on google:Google Search for Accessing ESRI Geodatabase with QGIS

You can also download esri's ArcGIS Explorer. It's free and will open the geodatabase.