This link explains the -IMEX- tables:
http://www.opengatesw.net/ms-access-tutorials/Access-Articles/Microsoft-Access-System-Tables.htm
I looked at the tables in one of my databases. They are just tables. Protected system tables ... but tables. There is no more a size limit (that I can see) than there is to any other table. You should be able to save hundreds or even thousands of such records. Unless you have an unusually long column name in one of the Excel column headers, I don't see where you would get that error. The column name is limited to 64 characters. But if I recall correctly, Excel also has that limit.
The structure APPEARS to be that mSysIMEXSpecs is the parent and mSysIMEXColumns is the child, with the ...Specs!SpecID as the linking field. The actual PK of the columns table is a compound of the spec ID number and the spreadsheet's column name.
Since SpecID is an Autonumber, I would expect that it would be largest for the most recent specs. (It isn't the PK, however - the spec's NAME is the PK.) So maybe you could open the table before an import to see what is there from prior imports and see if any of the numbers look exceeding large for some reason. Also check the highest-numbered entries in the ...IMEXSpecs table to see if any of the numbers look huge.
If you don't tell it to save, I wonder if there is some sort of improper cleanup going on? Since you say you are not saving specs, check for specs with a synthesized name, which you can recognize because the typical such name will start with a tilde (~) character. See if there ARE any, which should not be the case if you aren't saving.
I looked at database properties and offhand didn't see one relating to "always save import spec" or anything with a similar name, but I could have missed it.
As an extreme attempt, you MIGHT consider this a corrupted database and try to create a new database, importing your content from the one that is misbehaving.
Or you could try a different extremity: First make a copy of the database to protect yourself in case something goes bonkers on you. Then go into those tables and delete ALL ENTRIES from the mSysIMEXColumns and mSysIMEXSpecs tables (in that order, since they are child & parent so have Relational Integrity issues).
I have a database that has never done an import. For me, those tables are empty, so it doesn't seem to hurt anything that they ARE empty. If you don't have a "legit" import spec that you wanted to keep, that might help too. Unfortunately, if it is due to corruption, might NOT help a whit!