Possibly what's happening is a memory and disk resource limit. The Make-Table query on the N=350,000 records is expanding the database and doubling the size of existing data (i.e., in addition to its derived data). Remember number of rows is not the only factor in size: how wide the table (up to max of 255) and data types (double vs integers, memo vs text fields, etc.) impact database size as well. As this SO post shows, you can have a four-column integer table of 7 million records that comes under 200 MB!
Also, compared to the ordinary Select query, the Make-Table query runs the same full or select table scans multiple times depending on the number of joins (think Cartesian products on records) plus it extends to create a new database object, namely the table with all its definitions. This process can easily bloat and reach Access' 2GB limit or close to it slowing down the processing. It's possible if you stay long enough, the error message: Not enough space on temporary disk. (Error 3183) may appear.
Consider passing the Make Table into an external, empty or smaller Access database whose table can then be linked to current database:
SELECT * INTO [NewTable]
IN 'C:\Path\To\ExternalDatabase.accdb'
FROM [QueryName]
With automated VBA table link:
DoCmd.TransferDatabase acLink, "Microsoft Access", _
"C:\Path\To\ExternalDatabase.accdb", acTable, _
"SrcTableName", "DestTableName"
Alternatively, link a pre-existing external, empty/smaller Access table and run an append query inside current database:
INSERT INTO [LinkedTabe] SELECT * FROM [QueryName]