After reading what the error code means thanks to the answer from VonC I understood where to look. The problem was a lot more obscure that it seems.
I looked into the configuration folder for Eclipse (logs are either written there or in the .metadata folder when something goes wrong), and I found a huge log file. Inside the file I found the following error:
application org.eclipse.sdk not found
and the following exception, followed by a plugin name, several times:
java.util.zip.ZipException: Too many open files
Several plugins could not be loaded, causing a cascade of missing dependencies that prevented Eclipse from launching. I searched the web for this exception and found the following bug description from SUN, which says that Java 1.5.0 can not open more than 2,100 zip files.
The problem started a while after I installed the BABEL project translations for Eclipse into the build computer. These are more than 900 fragments, containing translations for many plugins, one for each language. As I installed it on top of an existing eclipse installation, it seemed that it was not a problem to open them.... until I cleared the workspace for the builds. Then Eclipse wouldn't launch anymore. I went over the limit. It didn't help that the first thing I tried to fix the build was, again.... clearing everything.
Because I only use this computer for headless builds, I didn't realize that the problem was in Eclipse itself and I was looking inside the build process. I only realized when I looked into the log file.
After installing Java 1.6.0_11 I was able to launch Eclipse and go on with my build.