Auto adjustment of table size automatically in ATLAS from QGIS

Unfortunately there isn't a setting to set the width of the cell based on the length of the text, so instead I'd suggest a couple of options:

First - under the 'Position and Size' properties, set the reference point to the right-hand position - this way, if text does force the table to become wider, it won't bleed off the page:

Second - set the 'Oversized Text' option to 'wrap text', that way it will limit how much the table will widen if text is too big.