Yes, disk space can cause a system crash. It's a dangerous situation that should be avoided. Processes can run out of space, log files can fill up, etc. Certain applications and daemons (e.g MySQL) will halt or crash if they run out of disk space.
In your case, it seems like your system doesn't have any partitions. You should provide the output of
df -h to start...
From there, you may want to see which directories are consuming the greatest amount of space:
du -skh /* should provide an accounting of which top-level directories are the largest. I prefer the ncdu utility for this, but you can't expect that to be installed on most systems.
du -skh /*, you'll get an output like this:
[root@xx /]# du -skh /*
If you wanted to drill down into
/var, for instance,
cd /var and run
du -skh * and look for the largest consumers of space... But of course, the process of discovering this has been covered numerous times on this site. Check these related questions for more suggestions:
How does one find which files are taking up 80% of the space on a Linux webserver?
How do I find out what is using up all the space on my / partition?
Disk usage treemap software for headless Linux