Hi alxc48, and welcome here!
Could you please post the start of BOINC's message log to allow us to see what your storage settings are?
There can be various reasons for this message, from really insufficient disk storage allocation to combined settings resulting into much less than what you think you allowed. The log clearly says which size BOINC "thinks" it may use.

Another reason is if the data directory of BOINC is shared with files from other applications. BOINC counts all file sizes in its data directory as BOINC files. For example if you store a few movies in BOINC's data directory (or in a subdirectory of its data directory) BOINC will probably find it is exceeding what it is allowed to use.

Otherwise, to answer your question, the data you crunch has to be stored on your PC from the time you received the job until its result files have been uploaded. But once this job has been reported as complete all related files are cleaned from your disk.

And thank you for having joined WCG.