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Will I be able to login into the system if the root filesystem is 100% full?
Configuration: home is not a partition but is also placed on the root (i.e. /home is a directory on the partition holding root /), var and tmp are separate partitions and in good state.
You should be able to log in as root, because usually a percentage of the partition's size is reserved in order to always enable root login for rescue operations and such. See this U&L Q&A:
Reserved space for root on a filesystem - why?
What you won't be able to do, however, is log in as a regular user from your display manager then switch to root or use sudo from a shell in a terminal.
You have two alternatives instead:
This assumes that the reason your partition was filled up is due to regular user activity and not activity by root processes. In such cases, you might need to resort to mounting the partition on a Live system and freeing up the space from there. Thanks to Alexios' comment for bringing this up.
root or a daemon running as the super user that filled up the filesystem in the first place. In that case, a single-user boot (or in extreme circumstances, rescue boot) may be necessary.