Try booting into Safe Mode.
If that also does not work, then boot into Recovery Mode. Launch Disk Utility and select the internal drive's Data volume. See what the "Free" space value is reported. There should be at least 35GB+ of Free space listed....more is better since it is hard to say how much the installer may need, plus it is hard to say what services you may have running in macOS that may be eating up storage space. By default the Monterey installer needs at least 35GB+ of Free space (or about 50GB+ if installing from Recovery Mode or USB stick).
You don't mention the exact model Mac, but perhaps the internal hard drive is failing. An OS install stresses a hard drive much more than regular use and utilizes portions of the drive that may rarely be used so it is not uncommon to encounter a drive failure during an OS install/upgrade. You can try running the Apple Diagnostics to see if any hardware issues are detected, but unfortunately the diagnostics don't usually detect problems...never hurts to try though.
Edit: Also disconnect all external devices in case one of them is causing a problem.