OR_DEM_10M.gdb is a folder, but it is the “file” that ArcMap or ArcCatolog will read.  It contains many files …. 

See the list here:
ftp://ftp.gis.oregon.gov/elevation/DEM/Statewide_Filegeodatabase/OR_DEM_10M.gdb/

One is 9 GB … ftp://ftp.gis.oregon.gov/elevation/DEM/Statewide_Filegeodatabase/OR_DEM_10M.gdb/a00000029.gdbtable

Use a good FTP utility like WINSCP. Point it to ftp://ftp.gis.oregon.gov/elevation/DEM/Statewide_Filegeodatabase/ 
Download OR_DEM_10M.gdb and it will grab all the component files in the subdirectory.

WINSCP instructions:
> New Site

LOGIN TAB
File Protocol: SFTP
Host: ftp.gis.oregon.gov
Port: 22
User:   anonymous  
Pa55:   yourname@youremailaddress.com 

Then "Login"

You may be prompted to accept an SSH certificate from our server.  Accept it

You need all the contents of the directory.  If you don’t have a folder named OR_DEM_10M.gdb with all the component files in that directory, your software will not recognize it as a Geodatabase.

Other option, if you don’t need the actual file you can point ArcMap or your software to our REST service … 
http://navigator.state.or.us/ArcGIS/rest/services/Framework/Elevation_DEM_10m/ImageServer